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The second-order ordinarydifferential equation(1)The solutions are the modified Bessel functions of the first and second kinds, and can be written(2)(3)where is a Bessel function of the first kind, is a Bessel function of the second kind, is a modified Bessel function of the first kind, and is modified Bessel function of the second kind.If , the modified Bessel differential equation becomes(4)which can also be written(5)

Underdamped simple harmonic motion is a special case of dampedsimple harmonic motion(1)in which(2)Since we have(3)it follows that the quantity(4)(5)is positive. Plugging in the trial solution to the differential equation then gives solutions that satisfy(6)i.e., the solutions are of the form(7)Using the Euler formula(8)this can be rewritten(9)We are interested in the real solutions. Since we are dealing here with a linear homogeneous ODE, linear sums of linearly independent solutions are also solutions. Since we have a sum of such solutions in (9), it follows that the imaginary and real parts separately satisfy the ODE and are therefore the solutions we seek. The constant in front of the sine term is arbitrary, so we can identify the solutions as(10)(11)so the general solution is(12)The initial values are(13)(14)so and can be expressed in terms of the initial conditions by(15)(16)The above plot shows an underdamped simple harmonic..

The ordinary differential equation(1)It has solution(2)where(3)and is a modified Bessel function of the first kind.

A method for solving ordinary differential equations using the formulawhich advances a solution from to . Note that the method increments a solution through an interval while using derivative information from only the beginning of the interval. As a result, the step's error is . This method is called simply "the Euler method" by Press et al. (1992), although it is actually the forward version of the analogous Euler backward method.While Press et al. (1992) describe the method as neither very accurate nor very stable when compared to other methods using the same step size, the accuracy is actually not too bad and the stability turns out to be reasonable as long as the so-called Courant-Friedrichs-Lewy condition is fulfilled. This condition states that, given a space discretization, a time step bigger than some computable quantity should not be taken. In situations where this limitation is acceptable, Euler's forward method becomes..

The general nonhomogeneous differential equation is given by(1)and the homogeneous equation is(2)(3)Now attempt to convert the equation from(4)to one with constant coefficients(5)by using the standard transformation for linear second-order ordinary differential equations. Comparing (3) and (5), the functions and are(6)(7)Let and define(8)(9)(10)(11)Then is given by(12)(13)(14)which is a constant. Therefore, the equation becomes a second-orderordinary differential equation with constant coefficients(15)Define(16)(17)(18)(19)and(20)(21)The solutions are(22)In terms of the original variable ,(23)Zwillinger (1997, p. 120) gives two other types of equations known as Euler differential equations,(24)(Valiron 1950, p. 201) and(25)(Valiron 1950, p. 212), the latter of which can be solved in terms of Bessel functions...

The partial differential equationwhich can also be rewritten

A method which can be used to solve the initial value problem for certain classes of nonlinear partial differential equations. The method reduces the initial value problem to a linear integral equation in which time appears only implicitly. However, the solutions and various of their derivatives must approach zero as (Infeld and Rowlands 2000).

The Benney equation in dimensions is the nonlinear partial differential equation

Consider the differential equation satisfied by(1)where is a Whittaker function, which is given by(2)(3)(Moon and Spencer 1961, p. 153; Zwillinger 1997, p. 128). This is usually rewritten(4)The solutions are parabolic cylinder functions.The equations(5)(6)which arise by separating variables in Laplace's equation in parabolic cylindrical coordinates, are also known as the Weber differential equations. As above, the solutions are known as parabolic cylinder functions.Zwillinger (1997, p. 127) calls(7)the Weber differential equation (Gradshteyn and Ryzhik 2000, p. 989).

If is an ordinary point of the ordinary differential equation, expand in a Taylor series about . Commonly, the expansion point can be taken as , resulting in the Maclaurin series(1)Plug back into the ODE and group the coefficients by power. Now, obtain a recurrence relation for the th term, and write the series expansion in terms of the s. Expansions for the first few derivatives are(2)(3)(4)(5)(6)If is a regular singular point of the ordinary differential equation,(7)solutions may be found by the Frobenius method or by expansion in a Laurent series. In the Frobenius method, assume a solution of the form(8)so that(9)(10)(11)(12)Now, plug back into the ODE and group the coefficients by power to obtain a recursion formula for the th term, and then write the series expansion in terms of the s. Equating the term to 0 will produce the so-called indicial equation, which will give the allowed values of in the series expansion.As an example, consider the..

Solution of a system of second-order homogeneous ordinary differential equations with constant coefficients of the formwhere is a positive definite matrix. To solve the vibration problem, 1. Solve the characteristic equation of to get eigenvalues , ..., . Define . 2. Compute the corresponding eigenvectors , ..., . 3. The normal modes of oscillation are given by , ..., , where , ..., and , ..., are arbitrary constants. 4. The general solution is .

Let be a real or complex piecewise-continuous function defined for all values of the real variable and that is periodic with minimum period so that(1)Then the differential equation(2)has two continuously differentiable solutions and , and the characteristic equation is(3)with eigenvalues and . Then Floquet's theorem states that if the roots and are different from each other, then (2) has two linearly independent solutions(4)(5)where and are periodic with period (Magnus and Winkler 1979, p. 4).

For a second-order ordinarydifferential equation,(1)Assume that linearly independent solutions and are known to the homogeneous equation(2)and seek and such that(3)(4)Now, impose the additional condition that(5)so that(6)(7)Plug , , and back into the original equation to obtain(8)which simplifies to(9)Combing equations (◇) and (9) and simultaneously solving for and then gives(10)(11)where(12)is the Wronskian, which is a function of only, so these can be integrated directly to obtain(13)(14)which can be plugged in to give the particular solution(15)Generalizing to an th degree ODE, let , ..., be the solutions to the homogeneous ODE and let , ..., be chosen such that(16)and the particular solution is then(17)

The modified spherical Bessel differential equation is given by the spherical Bessel differential equation with a negative separation constant,The solutions are called modified spherical Bessel functions of the first and second kinds.

Given a system of ordinary differentialequations of the form(1)that are periodic in , the solution can be written as a linear combination of functions of the form(2)where is a function periodic with the same period as the equations themselves. Given an ordinary differential equation of the form(3)where is periodic with period , the ODE has a pair of independent solutions given by the real and imaginary parts of(4)(5)(6)(7)Plugging these into (◇) gives(8)so the real and imaginaryparts are(9)(10)From (◇),(11)(12)(13)Integrating gives(14)where is a constant which must equal 1, so is given by(15)The real solution is then(16)so(17)(18)(19)(20)and(21)(22)(23)(24)which is an integral of motion. Therefore, although is not explicitly known, an integral always exists. Plugging (◇) into (◇) gives(25)which, however, is not any easier to solve than (◇)...

The van der Pol equation is an ordinary differential equation that can be derived from the Rayleigh differential equation by differentiating and setting . It is an equation describing self-sustaining oscillations in which energy is fed into small oscillations and removed from large oscillations. This equation arises in the study of circuits containing vacuum tubes and is given byIf , the equation reduces to the equation of simple harmonic motion

A Lyapunov function is a scalar function defined on a region that is continuous, positive definite, for all ), and has continuous first-order partial derivatives at every point of . The derivative of with respect to the system , written as is defined as the dot product(1)The existence of a Lyapunov function for which on some region containing the origin, guarantees the stability of the zero solution of , while the existence of a Lyapunov function for which is negative definite on some region containing the origin guarantees the asymptotical stability of the zero solution of .For example, given the system(2)(3)and the Lyapunov function , we obtain(4)which is nonincreasing on every region containing the origin, and thus the zero solution is stable.

The second-order ordinarydifferential equation(1)whose solutions may be written either(2)where is the repeated integral of the erfc function (Abramowitz and Stegun 1972, p. 299), or(3)where is a Hermite polynomial and is a confluent hypergeometric function of the first kind.

The ordinary differential equationwhere is a Jacobi elliptic function (Arscott 1981).

The ordinary differential equationwhere is the gamma function (Abramowitz and Stegun 1972, p. 496; Zwillinger 1997, p. 127). The solution iswhere and are Bessel functions of the first and second kinds, and is a Struve function (Abramowitz and Stegun 1972).

The Lommel differential equation is a generalization of the Besseldifferential equation given by(1)or, in the most general form, by(2)The case is the most common (Watson 1966, p. 345; Zwillinger 1997, p. 125; Gradshteyn and Ryzhik 2000, p. 937), and its solutions are given by(3)(4)where are Lommel functions. Note that is most commonly written simply as .The second-order ordinarydifferential equation(5)is sometimes also called the Lommel differential equation.

The second-order ordinarydifferential equation(1)is called Liouville's equation (Goldstein and Braun 1973; Zwillinger 1997, p. 124),as are the partial differential equations(2)(Matsumo 1987; Zwillinger 1997, p. 133) and(3)(Calogero and Degasperis 1982, p. 60; Zwillinger 1997, p. 133).

The most general forced form of the Duffing equation is(1)Depending on the parameters chosen, the equation can take a number of special forms. For example, with no damping and no forcing, and taking the plus sign, the equation becomes(2)(Bender and Orszag 1978, p. 547; Zwillinger 1997, p. 122). This equation can display chaotic behavior. For , the equation represents a "hard spring," and for , it represents a "soft spring." If , the phase portrait curves are closed.If instead we take , , reset the clock so that , and use the minus sign, the equation is then(3)This can be written as a system of first-order ordinary differential equations as(4)(5)(Wiggins 1990, p. 5) which, in the unforced case, reduces to(6)(7)(Wiggins 1990, p. 6; Ott 1993, p. 3).The fixed points of this set of coupled differential equations are given by(8)so , and(9)(10)giving . The fixed points are therefore , , and .Analysis..

Differential algebra is a field of mathematics that attempts to use methods from abstract algebra to study solutions of systems of polynomial nonlinear ordinary and partial differential equations. It is a generalization of classical commutative algebra and is primarily based on the work of Ritt (1950) and Kolchin (1973). Mansfield (1991) gave a terminating algorithm for differential Gröbner bases, which are differential analogs of polynomial Gröbner bases.

Take the Helmholtz differential equation(1)in spherical coordinates. This is just Laplace's equation in spherical coordinates with an additional term,(2)Multiply through by ,(3)This equation is separable in . Call the separation constant ,(4)Now multiply through by ,(5)This is the spherical Bessel differential equation. It can be transformed by letting , then(6)Similarly,(7)so the equation becomes(8)Now look for a solution of the form , denoting a derivative with respect to by a prime,(9)(10)(11)so(12)(13)(14)(15)(16)But the solutions to this equation are Bessel functionsof half integral order, so the normalized solutions to the original equation are(17)which are known as spherical Bessel functions. The two types of solutions are denoted (spherical Bessel function of the first kind) or (spherical Bessel function of the second kind), and the general solution is written(18)where(19)(20)..

The Legendre differential equation is the second-orderordinary differential equation(1)which can be rewritten(2)The above form is a special case of the so-called "associated Legendre differential equation" corresponding to the case . The Legendre differential equation has regular singular points at , 1, and .If the variable is replaced by , then the Legendre differential equation becomes(3)derived below for the associated () case.Since the Legendre differential equation is a second-order ordinary differential equation, it has two linearly independent solutions. A solution which is regular at finite points is called a Legendre function of the first kind, while a solution which is singular at is called a Legendre function of the second kind. If is an integer, the function of the first kind reduces to a polynomial known as the Legendre polynomial.The Legendre differential equation can be solved using the Frobenius method..

Given a simple harmonic oscillator with a quadratic perturbation, write the perturbation term in the form ,(1)find the first-order solution using a perturbation method. Write(2)and plug back into (1) and group powers to obtain(3)To solve this equation, keep terms only to order and note that, because this equation must hold for all powers of , we can separate it into the two simultaneous differential equations(4)(5)Setting our clock so that , the solution to (4) is then(6)Plugging this solution back into (5) then gives(7)The equation can be solved to give(8)Combining and then gives(9)(10)where the sinusoidal and cosinusoidal terms of order (from the ) have been ignored in comparison with the larger terms from .As can be seen in the top figure above, this solution approximates only for . As the lower figure shows, the differences from the unperturbed oscillator grow stronger over time for even relatively small values of ...

Simple harmonic motion refers to the periodic sinusoidal oscillation of an object or quantity. Simple harmonic motion is executed by any quantity obeying the differential equation(1)where denotes the second derivative of with respect to , and is the angular frequency of oscillation. This ordinary differential equation has an irregular singularity at . The general solution is(2)(3)where the two constants and (or and ) are determined from the initial conditions.Many physical systems undergoing small displacements, including any objects obeying Hooke's law, exhibit simple harmonic motion. This equation arises, for example, in the analysis of the flow of current in an electronic CL circuit (which contains a capacitor and an inductor). If a damping force such as Friction is present, an additional term must be added to the differential equation and motion dies out over time...

A generalization of the Bessel differentialequation for functions of order 0, given bySolutions arewhere is a confluent hypergeometric function of the first kind.

Adding a damping force proportional to to the equation of simple harmonic motion, the first derivative of with respect to time, the equation of motion for damped simple harmonic motion is(1)where is the damping constant. This equation arises, for example, in the analysis of the flow of current in an electronic CLR circuit, (which contains a capacitor, an inductor, and a resistor). The curve produced by two damped harmonic oscillators at right angles to each other is called a harmonograph, and simplifies to a Lissajous curve if .The damped harmonic oscillator can be solved by looking for trial solutions of the form . Plugging this into (1) gives(2)(3)This is a quadratic equation with solutions(4)There are therefore three solution regimes depending on the signof the quantity inside the square root,(5)The three regimes are summarized in the following table.regimeunderdampingcritical dampingoverdampingIf a periodic (sinusoidal) forcing..

Separation of variables is a method of solving ordinary and partial differential equations.For an ordinary differential equation(1)where is nonzero in a neighborhood of the initial value, the solution is given implicitly by(2)If the integrals can be done in closed form and the resulting equation can be solved for (which are two pretty big "if"s), then a complete solution to the problem has been obtained. The most important equation for which this technique applies is , the equation for exponential growth and decay (Stewart 2001).For a partial differential equation in a function and variables , , ..., separation of variables can be applied by making a substitution of the form(3)breaking the resulting equation into a set of independent ordinary differential equations, solving these for , , ..., and then plugging them back into the original equation.This technique works because if the product of functions of independent variables..

Whittaker and Watson (1990, pp. 539-540) write Lamé's differential equation for ellipsoidal harmonics of the first kind of the four types as(1)(2)(3)(4)where(5)(6)

The ordinary differential equation(1)(Byerly 1959, p. 255). The solution is denoted and is known as an ellipsoidal harmonic of the first kind, or Lamé function. Whittaker and Watson (1990, pp. 554-555) give the alternative forms (2)(3)(4)(5)(6)(Whittaker and Watson 1990, pp. 554-555; Ward 1987; Zwillinger 1997, p. 124). Here, is a Weierstrass elliptic function, is a Jacobi elliptic function, and(7)(8)(9)Two other equations named after Lamé are given by(10)and(11)(Moon and Spencer 1961, p. 157; Zwillinger 1997, p. 124).

The Laguerre differential equation is given by(1)Equation (1) is a special case of the more general associatedLaguerre differential equation, defined by(2)where and are real numbers (Iyanaga and Kawada 1980, p. 1481; Zwillinger 1997, p. 124) with .The general solution to the associated equation (2) is(3)where is a confluent hypergeometric function of the first kind and is a generalized Laguerre polynomial.Note that in the special case , the associated Laguerre differential equation is of the form(4)so the solution can be found using an integratingfactor(5)(6)(7)(8)as(9)(10)(11)where is the En-function.The associated Laguerre differential equation has a regular singular point at 0 and an irregular singularity at . It can be solved using a series expansion, (12)(13)(14)(15)(16)This requires(17)(18)for . Therefore,(19)for , 2, ..., so(20)(21)(22)If is a nonnegative integer, then the series terminates and the..

The method of d'Alembert provides a solution to the one-dimensional waveequation(1)that models vibrations of a string.The general solution can be obtained by introducing new variables and , and applying the chain rule to obtain(2)(3)(4)(5)Using (4) and (5) to compute the left andright sides of (3) then gives(6)(7)(8)(9)respectively, so plugging in and expanding then gives(10)This partial differential equation has general solution(11)(12)where and are arbitrary functions, with representing a right-traveling wave and a left-traveling wave.The initial value problem for a string located at position as a function of distance along the string and vertical speed can be found as follows. From the initial condition and (12),(13)Taking the derivative with respect to then gives(14)(15)and integrating gives(16)Solving (13) and (16) simultaneously for and immediately gives(17)(18)so plugging these into (13) then gives the solution..

The partial differential equation(Gray 1997, p. 399), whose solutions are called minimal surfaces. This corresponds to the mean curvature equalling 0 over the surface.d'Alembert's equationis sometimes also known as Lagrange's equation (Zwillinger 1997, pp. 120 and 265-268).

The ordinary differential equationwhere and and are given functions. This equation is sometimes also known as Lagrange's equation (Zwillinger 1997).

Critical damping is a special case of dampedsimple harmonic motion(1)in which(2)where is the damping constant. Therefore(3)In this case, so the solutions of the form satisfy(4)One of the solutions is therefore(5)In order to find the other linearly independent solution, we can make use of the identity(6)Since we have , simplifies to . Equation (6) therefore becomes(7)The general solution is therefore(8)In terms of the constants and , the initial values are(9)(10)so(11)(12)The above plot shows a critically damped simple harmonic oscillator with , for a variety of initial conditions .For sinusoidally forced simple harmonic motion with critical damping, the equation of motion is(13)and the Wronskian is(14)(15)Plugging this into the equation for the particular solution gives(16)(17)Applying the harmonic addition theoremthen gives(18)where(19)..

The system of ordinary differentialequations(1)(2)

The second-order ordinarydifferential equationsometimes also called Kummer's differential equation (Slater 1960, p. 2; Zwillinger 1997, p. 124). It has a regular singular point at 0 and an irregular singularity at . The solutionsare called confluent hypergeometric function of the first and second kinds, respectively. Note that the confluent hypergeometric function of the first kind is also denoted or .

Poisson's equation is(1)where is often called a potential function and a density function, so the differential operator in this case is . As usual, we are looking for a Green's function such that(2)But from Laplacian,(3)so(4)and the solution is(5)Expanding in the spherical harmonics gives(6)where and are greater than/less than symbols. this expression simplifies to(7)where are Legendre polynomials, and . Equations (6) and (7) give the addition theorem for Legendre polynomials.In cylindrical coordinates, the Green'sfunction is much more complicated,(8)where and are modified Bessel functions of the first and second kinds (Arfken 1985).

The inhomogeneous Helmholtz differentialequation is(1)where the Helmholtz operator is defined as . The Green's function is then defined by(2)Define the basis functions as the solutions to the homogeneous Helmholtz differential equation(3)The Green's function can then be expanded in terms of the s,(4)and the delta function as(5)Plugging (◇) and (◇) into (◇) gives(6)Using (◇) gives(7)(8)This equation must hold true for each , so(9)(10)and (◇) can be written(11)The general solution to (◇) is therefore(12)(13)

Generally speaking, a Green's function is an integral kernel that can be used to solve differential equations from a large number of families including simpler examples such as ordinary differential equations with initial or boundary value conditions, as well as more difficult examples such as inhomogeneous partial differential equations (PDE) with boundary conditions. Important for a number of reasons, Green's functions allow for visual interpretations of the actions associated to a source of force or to a charge concentrated at a point (Qin 2014), thus making them particularly useful in areas of applied mathematics. In particular, Green's function methods are widely used in, e.g., physics, and engineering.More precisely, given a linear differential operator acting on the collection of distributions over a subset of some Euclidean space , a Green's function at the point corresponding to is any solution of(1)where denotes the delta..

A differential equation is an equation that involves the derivatives of a function as well as the function itself. If partial derivatives are involved, the equation is called a partial differential equation; if only ordinary derivatives are present, the equation is called an ordinary differential equation. Differential equations play an extremely important and useful role in applied math, engineering, and physics, and much mathematical and numerical machinery has been developed for the solution of differential equations.

A universal differential equation (UDE) is a nontrivial differential-algebraic equation with the property that its solutions approximate to arbitrary accuracy any continuous function on any interval of the real line.Rubel (1981) found the first known UDE by showing that, given any continuous function and any positive continuous function , there exists a solution of(1)such that(2)for all .Duffin (1981) found two additional families of UDEs,(3)and(4)whose solutions are for .Briggs (2002) found a further family of UDEs given by(5)for .

A delay differential equation (also called a differential delay equation or difference-differential equation, although the latter term has a different meaning in the modern literature) is a special type of functional differential equation. Delay differential equations are similar to ordinary differential equations, but their evolution involves past values of the state variable. The solution of delay differential equations therefore requires knowledge of not only the current state, but also of the state a certain time previously.Examples include the equations defining the Dickmanfunction(1)and the Buchstab function(2)(Panario 1998).

The Buchstab function is defined by the delay differential equation(1)(Panario 1998). It approaches the asymptotic value as (and in fact has nearly reached this value already by ).

Consider a second-orderordinary differential equationIf and remain finite at , then is called an ordinary point. If either or diverges as , then is called a singular point. If either or diverges as but and remain finite as , then is called a regular singular point (or nonessential singularity).

The complex second-orderordinary differential equation(1)(Abramowitz and Stegun 1972, p. 379; Zwillinger 1997, p. 123), whose solutions can be given in terms of the Kelvin functions as(2)(3)(4)(5)(Abramowitz and Stegun 1972, p. 379).The general solution is(6)where is a Bessel function of the first kind and is a Bessel function of the second kind.

(1)or(2)where is a function of one variable and . The general solution is(3)The singular solution envelopes are and .A partial differential equation knownas Clairaut's equation is given by(4)(Iyanaga and Kawada 1980, p. 1446; Zwillinger 1997, p. 132).

(1)for . The Chebyshev differential equation has regular singular points at , 1, and . It can be solved by series solution using the expansions(2)(3)(4)(5)(6)(7)(8)Now, plug equations (6) and (8) into theoriginal equation (◇) to obtain(9)(10)(11)(12)(13)so(14)(15)and by induction,(16)for , 3, ....Since (14) and (15) are special cases of (16), the general recurrence relation can be written(17)for , 1, .... From this, we obtain for the even coefficients(18)(19)(20)and for the odd coefficients(21)(22)(23)The even coefficients can be given in closed form as(24)(25)and the odd coefficients as(26)(27)The general solution is then given by summing over all indices,(28)which can be done in closed form as(29)Performing a change of variables gives the equivalent form of the solution(30)(31)where is a Chebyshev polynomial of the first kind and is a Chebyshev polynomial of the second kind. Another equivalent form of the solution..

where . Differentiating and setting gives the van der Pol equation. The equationwith the replaced by 1 is sometimes also called the Rayleigh differential equation (Birkhoff and Rota 1978, p. 134; Zwillinger 1997, p. 126).

Consider a second-orderordinary differential equationIf and remain finite at , then is called an ordinary point. If either or diverges as , then is called a singular point. If diverges more quickly than , so approaches infinity as , or diverges more quickly than so that goes to infinity as , then is called an irregular singularity (or essential singularity).

If is an analytic function in a neighborhood of the point (i.e., it can be expanded in a series of nonnegative integer powers of and ), find a solution of the differential equationwith initial conditions and . The existence and uniqueness of the solution were proven by Cauchy and Kovalevskaya in the Cauchy-Kovalevskaya theorem. The Cauchy problem amounts to determining the shape of the boundary and type of equation which yield unique and reasonable solutions for the Cauchy conditions.

A diagram used in the solution of ordinary differential equations ofthe formwhich vanish when , where(Ince 1956, pp. 298 and 427). The diagram is named in order of French mathematician Victor Puiseux.

An integrating factor is a function by which an ordinary differential equation can be multiplied in order to make it integrable. For example, a linear first-order ordinary differential equation of type(1)where and are given continuous functions, can be made integrable by letting be a function such that(2)and(3)Then would be the integrating factor such that multiplying by gives the expression(4)(5)using the product rule. Integrating both sides with respect to then gives the solution(6)

An initial value problem is a problem that has its conditions specified at some time . Usually, the problem is an ordinary differential equation or a partial differential equation. For example,(1)where denotes the boundary of , is an initial value problem.

If is a continuous function that satisfies the Lipschitz condition(1)in a surrounding of , then the differential equation(2)(3)has a unique solution in the interval , where , min denotes the minimum, , and sup denotes the supremum.

It has regular singular points at 0, 1, and . Every second-order ordinary differential equation with at most three regular singular points can be transformed into the hypergeometric differential equation.

A boundary value problem is a problem, typically an ordinary differential equation or a partial differential equation, which has values assigned on the physical boundary of the domain in which the problem is specified. For example,(1)where denotes the boundary of , is a boundary problem.

A linear ordinary differential equation of order is said to be homogeneous if it is of the form(1)where , i.e., if all the terms are proportional to a derivative of (or itself) and there is no term that contains a function of alone.However, there is also another entirely different meaning for a first-order ordinary differential equation. Such an equation is said to be homogeneous if it can be written in the form(2)Such equations can be solved in closed form by the change of variables which transforms the equation into the separable equation(3)

The second-order ordinarydifferential equation(1)where are fixed constants. A general solution can be given by taking the "determinant" of an infinite matrix.If only the term is present, the equation have solution(2)If terms are included, the equation becomes the Mathieu differential equation, which has solution(3)If terms are included, it becomes the Whittaker-Hill differential equation.

The Bessel differential equation is the linear second-orderordinary differential equation given by(1)Equivalently, dividing through by ,(2)The solutions to this equation define the Bessel functions and . The equation has a regular singularity at 0 and an irregular singularity at .A transformed version of the Bessel differential equation given by Bowman (1958) is(3)The solution is(4)where(5) and are the Bessel functions of the first and second kinds, and and are constants. Another form is given by letting , , and (Bowman 1958, p. 117), then(6)The solution is(7)

(1)Let for . Then(2)Rewriting (1) gives(3)(4)Plugging (4) into (3),(5)Now, this is a linear first-orderordinary differential equation of the form(6)where and . It can therefore be solved analytically using an integrating factor(7)(8)where is a constant of integration. If , then equation (◇) becomes(9)(10)(11)The general solution is then, with and constants,(12)

The second-order ordinarydifferential equation(1)This differential equation has an irregular singularity at . It can be solved using the series method(2)(3)Therefore,(4)and(5)for , 2, .... Since (4) is just a special case of (5),(6)for , 1, ....The linearly independent solutions are then(7)(8)These can be done in closed form as(9)(10)where is a confluent hypergeometric function of the first kind and is a Hermite polynomial. In particular, for , 2, 4, ..., the solutions can be written(11)(12)(13)where is the erfi function.If , then Hermite's differential equation becomes(14)which is of the form and so has solution(15)(16)(17)

The parabolic cylinder differential equation is the second-orderordinary differential equation(1)whose solution is given by(2)where is a parabolic cylinder function.The generalized parabolic cylinder differential equation is the differential equation of the form(3)(Abramowitz and Stegun 1972, p. 686; Zwillinger 1995, p. 414; Zwillinger 1997, p. 126) whose solution can be expressed in terms of parabolic cylinder functions as(4)where(5)

A differential equation or system of ordinary differential equations is said to be autonomous if it does not explicitly contain the independent variable (usually denoted ). A second-order autonomous differential equation is of the form , where . By the chain rule, can be expressed asFor an autonomous ODE, the solution is independent of the time at which the initial conditions are applied. This means that all particles pass through a given point in phase space. A nonautonomous system of first-order ODEs can be written as an autonomous system of ODEs by letting and increasing the dimension of the system by 1 by adding the equation

There are six Painlevé transcendents, corresponding to second-order ordinary differential equations whose only movable singularities are ordinary poles and which cannot be integrated in terms of other known functions or transcendents.(1)(2)(3)(4)(5)(6)(Painlevé 1906; Ince 1956, p. 345; Zwillinger 1997, pp. 125-126).All Painlevé transcendents have first integrals for special values of their parameters except equation (2). Five of the transcendents were found by Painlevé and his students; the sixth transcendent was found by Fuchs (1905, 1907; Hille 1997, p. 440) and contains the other five as limiting cases (Garnier 1916ab; Ince 1956, p. 345).The sixth Painlevé transcendent is one of the most important nonlinear differential equations for defining new transcendental functions...

The study, first developed by Boole, of shift-invariant operators which are polynomials in the differential operator . Heaviside calculus can be used to solve any ordinary differential equation of the formwith , and is frequently implemented using Laplace transforms.

The associated Legendre differential equation is a generalization of the Legendredifferential equation given by(1)which can be written(2)(Abramowitz and Stegun 1972; Zwillinger 1997, p. 124). The solutions to this equation are called the associated Legendre polynomials (if is an integer), or associated Legendre functions of the first kind (if is not an integer). The complete solution is(3)where is a Legendre function of the second kind.The associated Legendre differential equation is often written in a form obtained by setting . Plugging the identities(4)(5)(6)(7)into (◇) then gives(8)(9)

Following the work of Fuchs in classifying first-order ordinary differential equations, Painlevé studied second-order ordinary differential equation of the formwhere is analytic in and rational in and . Painlevé found 50 types whose only movable singularities are ordinary poles. This characteristic is known as the Painlevé property. Six of the transcendents define new transcendents known as Painlevé transcendents, and the remaining 44 can be integrated in terms of classical transcendents, quadratures, or the Painlevé transcendents.

Overdamped simple harmonic motion is a special case of dampedsimple harmonic motion(1)in which(2)Therefore(3)(4)(5)where(6)The general solution is therefore(7)where and are constants. The initial values are(8)(9)so(10)(11)The above plot shows an overdamped simple harmonic oscillator with , and three different initial conditions .For a cosinusoidally forced overdamped oscillator with forcing function , i.e.,(12)the general solutions are(13)(14)where(15)(16)These give the identities(17)(18)and(19)(20)We can now use variation of parametersto obtain the particular solution as(21)where(22)(23)and the Wronskian is(24)(25)These can be integrated directly to give(26)(27)Integrating, plugging in, and simplifying then gives(28)(29)where use has been made of the harmonic additiontheorem and(30)

Some authors define a general Airy differential equation as(1)This equation can be solved by series solution using the expansions(2)(3)(4)(5)(6)(7)(8)Specializing to the "conventional" Airy differential equation occurs by taking the minus sign and setting . Then plug (8) into(9)to obtain(10)(11)(12)(13)In order for this equality to hold for all , each term must separately be 0. Therefore,(14)(15)Starting with the term and using the above recurrence relation, we obtain(16)Continuing, it follows by induction that(17)for , 2, .... Now examine terms of the form .(18)(19)(20)Again by induction,(21)for , 2, .... Finally, look at terms of the form ,(22)(23)(24)By induction,(25)for , 2, .... The general solution is therefore(26)For a general with a minus sign, equation (◇) is(27)and the solution is(28)where is a modified Bessel function of the first kind. This is usually expressed in terms of the Airy functions and..

The generalized hypergeometricfunctionsatisfies the equationwhere is the differential operator.

To solve the system of differential equations(1)where is a matrix and and are vectors, first consider the homogeneous case with . The solutions to(2)are given by(3)But, by the eigen decomposition theorem,the matrix exponential can be written as(4)where the eigenvector matrixis(5)and the eigenvalue matrixis(6)Now consider(7)(8)(9)The individual solutions are then(10)so the homogeneous solution is(11)where the s are arbitrary constants.The general procedure is therefore 1. Find the eigenvalues of the matrix (, ..., ) by solving the characteristic equation. 2. Determine the corresponding eigenvectors , ..., . 3. Compute(12)for , ..., . Then the vectors which are real are solutions to the homogeneous equation. If is a matrix, the complex vectors correspond to real solutions to the homogeneous equation given by and . 4. If the equation is nonhomogeneous, find the particular solution given by(13)where the matrix is defined by(14)If..

A generalization of the confluenthypergeometric differential equation given by(1)The solutions are given by(2)(3)where is a confluent hypergeometric function of the first kind and is a confluent hypergeometric function of the second kind (Abramowitz and Stegun 1972, p. 505).

Given a homogeneous linear second-orderordinary differential equation,(1)call the two linearly independent solutions and . Then(2)(3)Now, take (3) minus (2),(4)Now, use the definition of the Wronskian and take itsderivative,(5)(6)(7)Plugging and into (4) gives(8)This can be rearranged to yield(9)which can then be directly integrated to(10)where is the natural logarithm. Exponentiating then yields Abel's identity(11)where is a constant of integration.

The second-order ordinarydifferential equation(1)sometimes called the hyperspherical differential equation (Iyanaga and Kawada 1980, p. 1480; Zwillinger 1997, p. 123). The solution to this equation is(2)where is an associated Legendre function of the first kind and is an associated Legendre function of the second kind.A number of other forms of this equation are sometimes also known as the ultraspherical or Gegenbauer differential equation, including(3)The general solutions to this equation are(4)If is an integer, then one of the solutions is known as a Gegenbauer polynomials , also known as ultraspherical polynomials.The form(5)is also given by Infeld and Hull (1951, pp. 21-68) and Zwillinger (1997, p. 122). It has the solution(6)

An integral equation ofthe formwhere is the integral kernel, is a specified function, and is the function to be solved for.

An integral equation ofthe formwhere is the integral kernel, is a specified function, and is the function to be solved for.

A Fredholm integral equationof the second kind(1)may be solved as follows. Take(2)(3)(4)(5)where(6)(7)(8)(9)The Neumann series solution is then(10)

An equation involving a function and integrals of that function to solved for . If the limits of the integral are fixed, an integral equation is called a Fredholm integral equation. If one limit is variable, it is called a Volterra integral equation. If the unknown function is only under the integral sign, the equation is said to be of the "first kind." If the function is both inside and outside, the equation is called of the "second kind." An example integral equation is given by(1)(Kress 1989, 1998), which has solution .Let be the function to be solved for, a given known function, and a known integral kernel. A Fredholm integral equation of the first kind is an integral equation of the form(2)A Fredholm integral equationof the second kind is an integral equation of the form(3)A Volterra integral equationof the first kind is an integral equation of the form(4)A Volterra integral equationof the second kind is an integral equation..

An integral equation ofthe form(1)(2)The solution to a general Fredholm integral equation of the second kind is calledan integral equation Neumann series.A Fredholm integral equation of the second kind with separable integralkernel may be solved as follows:(3)(4)(5)where(6)Now multiply both sides of (◇) by and integrate over .(7)By (◇), the first term is just . Now define(8)(9)so (◇) becomes(10)Writing this in matrix form,(11)so(12)(13)

A Fredholm integral equation of the first kind is an integralequation of the form(1)where is the kernel and is an unknown function to be solved for (Arfken 1985, p. 865).If the kernel is of the special form and the limits are infinite so that the equation becomes(2)then the solution (assuming the relevant transforms exist) is given by(3)where is the Fourier transforms operator (Arfken 1985, pp. 875 and 877).

The term "characteristic" has many different uses in mathematics. In general, it refers to some property that inherently describes a given mathematical object, for example characteristic class, characteristic equation, characteristic factor, etc. However, the unqualified term "characteristic" also has a number of specific meanings depending on context.For a real number , is called the characteristic, where is the floor function.A path in a two-dimensional plane used to transform partial differential equations into systems of ordinary differential equations is also called a characteristic. This form of characteristic was invented by Riemann. For an example of the use of characteristics, consider the equationNow let . Sinceit follows that , , and . Integrating gives , , and , where the constants of integration are 0 and ...

The word adjoint has a number of related meanings. In linear algebra, it refers to the conjugate transpose and is most commonly denoted . The analogous concept applied to an operator instead of a matrix, sometimes also known as the Hermitian conjugate (Griffiths 1987, p. 22), is most commonly denoted using dagger notation (Arfken 1985). The adjoint operator is very common in both Sturm-Liouville theory and quantum mechanics. For example, Dirac (1982, p. 26) denotes the adjoint of the bra vector as , or .Given a second-order ordinarydifferential equation(1)with differential operator(2)where and , the adjoint operator is defined by(3)(4)Writing the two linearly independent solutions as and , the adjoint operator can then also be written(5)(6)In general, given two adjoint operators and ,(7)which can be generalized to(8)..

(1)or(2)The solutions are Jacobi polynomials or, in terms of hypergeometric functions, as(3)The equation (2) can be transformed to(4)where(5)and(6)where(7)Zwillinger (1997, p. 123) gives a related differential equation he terms Jacobi's equation(8)(Iyanaga and Kawada 1980, p. 1480), which has solution(9)Zwillinger (1997, p. 120; duplicated twice) also gives another types of ordinary differential equation called a Jacobi equation,(10)(Ince 1956, p. 22).In the calculus of variations, the partialdifferential equation(11)where(12)is called the Jacobi differential equation.

The equations defined by(1)(2)where and is fluxion notation and is the so-called Hamiltonian, are called Hamilton's equations. These equations frequently arise in problems of celestial mechanics.The vector form of these equations is(3)(4)(Zwillinger 1997, p. 136; Iyanaga and Kawada 1980, p. 1005).Another formulation related to Hamilton's equation is(5)where is the so-called Lagrangian.

The system of partial differential equations(1)(2)(3)where denotes the commutator.

In cylindrical coordinates, the scale factors are , , , so the Laplacian is given by(1)Attempt separation of variables in theHelmholtz differential equation(2)by writing(3)then combining (1) and (2) gives(4)Now multiply by ,(5)so the equation has been separated. Since the solution must be periodic in from the definition of the circular cylindrical coordinate system, the solution to the second part of (5) must have a negative separation constant(6)which has a solution(7)Plugging (7) back into (5) gives(8)and dividing through by results in(9)The solution to the second part of (9) must not be sinusoidal at for a physical solution, so the differential equation has a positive separation constant(10)and the solution is(11)Plugging (11) back into (9) and multiplying through by yields(12)But this is just a modified form of the Besseldifferential equation, which has a solution(13)where and are Bessel functions of the first and second kinds,..

Spinor fields describing particles of zero rest mass satisfy the so-called zero rest mass equations. Examples of zero rest mass particles include the neutrino (a fermion) and the gauge bosons (as long as gauge symmetry is not violated) such as the photon.If is the spinor field describing a particle of spin (where upper case Latin indices are spinor indices which can take the values 0 and 1), then it is symmetric and has indices. If the particle is also of zero rest mass, then satisfies the zero rest mass equationHere, in a Lorentz transformation, primed spinors transform under the conjugate of the transformation for unprimed ones, Einstein summation is used throughout, and denotes the spinor, which is equivalent to the Levi-Civita connection on Minkowski space. has one index for the neutrino, two for the photon, and four for the graviton. For the photon, the equation obtained states the vanishing of the divergence of the field strength tensor...

In two-dimensional Cartesian coordinates,attempt separation of variables by writing(1)then the Helmholtz differential equationbecomes(2)Dividing both sides by gives(3)This leads to the two coupled ordinary differential equations with a separation constant ,(4)(5)where and could be interchanged depending on the boundary conditions. These have solutions(6)(7)(8)The general solution is then(9)In three-dimensional Cartesian coordinates,attempt separation of variables by writing(10)then the Helmholtz differential equationbecomes(11)Dividing both sides by gives(12)This leads to the three coupled differential equations(13)(14)(15)where , , and could be permuted depending on boundary conditions. The general solution is therefore(16)

A partial derivative of second or greater orderwith respect to two or more different variables, for exampleIf the mixed partial derivatives exist and are continuous at a point , then they are equal at regardless of the order in which they are taken.

Maxwell's equations are the system of partial differential equations describing classical electromagnetism and therefore of central importance in physics. In the so-called cgs system of units, Maxwell's equations are given by(1)(2)(3)(4)where is the electric displacement field, is the free charge density, is the electric field, is the speed of light, is the magnetic field, is the magnetizing field, and is the free current density (cf. Purcell 1985, p. 330, 381, and 432; Jackson 1998, p. 818). As usual, is the divergence and is the curl.In the MKS system of units, the equations are written(5)(6)(7)(8)supplemented by(9)(10)where is the permittivity of free space, is the permeability of free space, is the electric polarization, and is the magnetic polarization or "magnetization" (Griffiths 1998, pp. 278-279)...

An elliptic partial differentialequation given by(1)where is a scalar function and is the scalar Laplacian, or(2)where is a vector function and is the vector Laplacian (Moon and Spencer 1988, pp. 136-143).When , the Helmholtz differential equation reduces to Laplace's equation. When (i.e., for imaginary ), the equation becomes the space part of the diffusion equation.The Helmholtz differential equation can be solved by separation of variables in only 11 coordinate systems, 10 of which (with the exception of confocal paraboloidal coordinates) are particular cases of the confocal ellipsoidal system: Cartesian, confocal ellipsoidal, confocal paraboloidal, conical, cylindrical, elliptic cylindrical, oblate spheroidal, paraboloidal, parabolic cylindrical, prolate spheroidal, and spherical coordinates (Eisenhart 1934ab). Laplace's equation (the Helmholtz differential equation with ) is separable in the two additional..

To solve the heat conduction equation on a two-dimensional disk of radius , try to separate the equation using(1)Writing the and terms of the Laplacian in cylindrical coordinates gives(2)so the heat conduction equation becomes(3)Multiplying through by gives(4)The term can be separated.(5)which has a solution(6)The remaining portion becomes(7)Dividing by gives(8)where a negative separation constant has been chosen so that the portion remains finite(9)The radial portion then becomes(10)(11)which is the spherical Besseldifferential equation.Consider disk or radius with initial temperature and the boundary condition . Then the solution is(12)where is the th positive zero of the Bessel function of the first kind (Bowman 1958, pp. 37-39).

The equation of motion for a membrane shaped as a right isosceles triangle of length on a side and with the sides oriented along the positive and axes is given bywhereand , integers with . This solution can be obtained by subtracting two wave solutions for a square membrane with the indices reversed. Since points on the diagonal which are equidistant from the center must have the same wave equation solution (by symmetry), this procedure gives a wavefunction which will vanish along the diagonal as long as and are both even or odd. We must further restrict the modes since those with give wavefunctions which are just the negative of and give an identically zero wavefunction.The plots above show the lowest order spatial modes.

A pair of linear operators and associated with a given partial differential equation which can be used to solve the equation. However, it turns out to be very difficult to find the and corresponding to a given equation, so it is actually simpler to postulate a given and and determine to which partial differential equation they correspond (Infeld and Rowlands 2000).

A partial differential diffusionequation of the form(1)Physically, the equation commonly arises in situations where is the thermal diffusivity and the temperature.The one-dimensional heat conduction equation is(2)This can be solved by separation of variablesusing(3)Then(4)Dividing both sides by gives(5)where each side must be equal to a constant. Anticipating the exponential solution in , we have picked a negative separation constant so that the solution remains finite at all times and has units of length. The solution is(6)and the solution is(7)The general solution is then(8)(9)(10)If we are given the boundary conditions(11)and(12)then applying (11) to (10) gives(13)and applying (12) to (10) gives(14)so (10) becomes(15)Since the general solution can have any ,(16)Now, if we are given an initial condition , we have(17)Multiplying both sides by and integrating from 0 to gives(18)Using the orthogonality of and ,(19)(20)(21)so(22)If..

To find the motion of a rectangular membrane with sides of length and (in the absence of gravity), use the two-dimensional wave equation(1)where is the vertical displacement of a point on the membrane at position () and time . Use separation of variables to look for solutions of the form(2)Plugging (2) into (1) gives(3)where the partial derivatives have now become complete derivatives. Multiplying (3) by gives(4)The left and right sides must both be equal to a constant, so we can separate the equation by writing the right side as(5)This has solution(6)Plugging (5) back into (◇),(7)which we can rewrite as(8)since the left and right sides again must both be equal to a constant. We can now separate out the equation(9)where we have defined a new constant satisfying(10)Equations (◇) and (◇) have solutions(11)(12)We now apply the boundary conditions to (11) and (12). The conditions and mean that(13)Similarly, the conditions..

The Laplacian for a scalar function is a scalar differential operator defined by(1)where the are the scale factors of the coordinate system (Weinberg 1972, p. 109; Arfken 1985, p. 92).Note that the operator is commonly written as by mathematicians (Krantz 1999, p. 16).The Laplacian is extremely important in mechanics, electromagnetics, wave theory, and quantum mechanics, and appears in Laplace's equation(2)the Helmholtz differential equation(3)the wave equation(4)and the Schrödinger equation(5)The analogous operator obtained by generalizing from three dimensions to four-dimensional spacetime is denoted and is known as the d'Alembertian. A version of the Laplacian that operates on vector functions is known as the vector Laplacian, and a tensor Laplacian can be similarly defined. The square of the Laplacian is known as the biharmonic operator.A vector Laplacian can also be defined, as canits generalization..

In toroidal coordinates, Laplace'sequation becomes(1)Attempt separation of variables by pluggingin the trial solution(2)then divide the result by to obtain(3)The function then separates with(4)giving solution(5)Plugging back in and dividing by gives(6)The function then separates with(7)giving solution(8)Plugging back in and multiplying by gives(9)which can also be written(10)(Arfken 1970, pp. 114-115). Laplace's equation is partially separable, although the Helmholtz differential equation is not.Solutions to the differential equation for are known as toroidal functions.

For the hyperbolic partialdifferential equation(1)(2)(3)on a domain , Goursat's problem asks to find a solution of (3) from the boundary conditions(4)(5)(6)for that is regular in and continuous in the closure , where and are specified continuously differentiable functions.The linear Goursat problem corresponds to the solution of the equation(7)which can be effected using the so-called Riemann function . The use of the Riemann function to solve the linear Goursat problem is called the Riemann method.

In spherical coordinates, the scale factors are , , , and the separation functions are , , , giving a Stäckel determinant of .The Laplacian is(1)To solve Laplace's equation in spherical coordinates, attempt separation of variables by writing(2)Then the Helmholtz differential equationbecomes(3)Now divide by ,(4)(5)The solution to the second part of (5) must be sinusoidal, sothe differential equation is(6)which has solutions which may be defined either as a complex function with , ..., (7)or as a sum of real sine and cosine functions with , ..., (8)Plugging (6) back into (7),(9)The radial part must be equal to a constant(10)(11)But this is the Euler differential equation,so we try a series solution of the form(12)Then(13)(14)(15)This must hold true for all powers of . For the term (with ),(16)which is true only if and all other terms vanish. So for , . Therefore, the solution of the component is given by(17)Plugging (17) back into (◇),(18)(19)which..

The wave equation is the important partialdifferential equation(1)that describes propagation of waves with speed . The form above gives the wave equation in three-dimensional space where is the Laplacian, which can also be written(2)An even more compact form is given by(3)where is the d'Alembertian, which subsumes the second time derivative and second space derivatives into a single operator.The one-dimensional wave equation isgiven by(4)As with all partial differential equations, suitable initial and/or boundary conditions must be given to obtain solutions to the equation for particular geometries and starting conditions.

In conical coordinates, Laplace'sequation can be written(1)where(2)(3)(Byerly 1959). Letting(4)breaks (1) into the two equations,(5)(6)Solving these gives(7)(8)where are ellipsoidal harmonics. The regular solution is therefore(9)However, because of the cylindrical symmetry, the solution is an th degree spherical harmonic.

Using the notation of Byerly (1959, pp. 252-253),Laplace's equation can be reduced to(1)where(2)(3)(4)(5)(6)(7)In terms of , , and ,(8)(9)(10)Equation (◇) is not separable using a function ofthe form(11)but it is if we let(12)(13)(14)These give(15)(16)and all others terms vanish. Therefore (◇) can be broken up into the equations(17)(18)(19)For future convenience, now write(20)(21)then(22)(23)(24)Now replace , , and to obtain(25)Each of these is a Lamé's differential equation, whose solution is called an ellipsoidal harmonic of the first kind. Writing(26)(27)(28)gives the solution to (◇) as a product of ellipsoidal harmonics of the first kind .(29)

The system of partial differential equations(1)(2)where is the biharmonic operator.

In bispherical coordinates, Laplace'sequation becomes(1)Attempt separation of variables by pluggingin the trial solution(2)then divide the result by to obtain(3)The function then separates with(4)giving solution(5)Plugging back in and dividing by gives(6)The function then separates with(7)giving solution(8)Plugging back in and multiplying by gives(9)so Laplace's equation is partially separable in bispherical coordinates. However, the Helmholtz differential equation cannot be separated in this manner.

In two-dimensional bipolar coordinates, Laplace's equation iswhich simplifies toso Laplace's equation is separable, although the Helmholtz differential equation is not.

The scalar form of Laplace's equation is the partialdifferential equation(1)where is the Laplacian.Note that the operator is commonly written as by mathematicians (Krantz 1999, p. 16). Laplace's equation is a special case of the Helmholtz differential equation(2)with , or Poisson's equation(3)with .The vector Laplace's equation is given by(4)A function which satisfies Laplace's equation is said to be harmonic. A solution to Laplace's equation has the property that the average value over a spherical surface is equal to the value at the center of the sphere (Gauss's harmonic function theorem). Solutions have no local maxima or minima. Because Laplace's equation is linear, the superposition of any two solutions is also a solution.A solution to Laplace's equation is uniquely determined if (1) the value of the function is specified on all boundaries (Dirichlet boundary conditions) or (2) the normal derivative of the function is..

The system of partial differential equationsdescribing fluid flow in the absence of viscosity, given bywhere is the fluid velocity, is the pressure, and is the fluid density.

The partial differential equationwhere is the real part of (Calogero and Degasperis 1982, p. 62; Zwillinger 1997, p. 131).

A second-order partial differential equation,i.e., one of the form(1)is called elliptic if the matrix(2)is positive definite. Elliptic partial differential equations have applications in almost all areas of mathematics, from harmonic analysis to geometry to Lie theory, as well as numerous applications in physics. As with a general PDE, elliptic PDE may have non-constant coefficients and be non-linear. Despite this variety, the elliptic equations have a well-developed theory.The basic example of an elliptic partial differential equation is Laplace'sequation(3)in -dimensional Euclidean space, where the Laplacian is defined by(4)Other examples of elliptic equations include the nonhomogeneous Poisson'sequation(5)and the non-linear minimal surface equation.For an elliptic partial differential equation, boundary conditions are used to give the constraint on , where(6)holds in .One property of constant coefficient elliptic..

In three dimensions, the spherical harmonic differential equation is given byand solutions are called spherical harmonics (Zwillinger 1997, p. 130). In four dimensions, the spherical harmonic differential equation is(Humi 1987; Zwillinger 1997, p. 130).

The Einstein field equations are the 16 coupled hyperbolic-elliptic nonlinear partial differential equations that describe the gravitational effects produced by a given mass in general relativity. As result of the symmetry of and , the actual number of equations reduces to 10, although there are an additional four differential identities (the Bianchi identities) satisfied by , one for each coordinate.The Einstein field equations state thatwhere is the stress-energy tensor, andis the Einstein tensor, with the Ricci curvature tensor and the scalar curvature.The opening sequence of the 2003 French film Les Triplettes de Belleville (The Triplets of Belleville) features the Einstein field equations.

The quantum electrodynamical law which applies to spin-1/2 particles and is the relativistic generalization of the Schrödinger equation. In dimensions (three space dimensions and one time dimension), it is given by(1)(Bjorken and Drell 1964, p. 6), where is h-bar, is the speed of light, is the wavefunction, is the mass of the particle, and are the Dirac matrices (with being called by Bjorken and Drell 1964, p. 8; Berestetskii et al. 1982, p. 78).The Dirac equation can also be written in the concise form(2)(Griffiths 1987, p. 216), where(3) are Dirac matrices in the "Dirac basis" (Griffiths 1987, p. 216), and Einstein summation has been used to sum over , 1, 2, 3.In dimensions, a generalization of the Dirac equation is given by the system of partial differential equations(4)(5)(Alvarez et al. 1982; Zwillinger 1997, p. 137), where corresponds to the quantum electrodynamical equation...

The Schrödinger equation describes the motion of particles in nonrelativistic quantum mechanics, and was first written down by Erwin Schrödinger. The time-dependent Schrödinger equation is given by(1)where is the reduced Planck constant , is the time-dependent wavefunction, is the mass of a particle, is the Laplacian, is the potential, and is the Hamiltonian operator. The time-independent Schrödinger equation is(2)where is the energy of the particle.The one-dimensional versions of these equations are then(3)and(4)Variants of the one-dimensional Schrödinger equation have been considered in various contexts, including the following (where is a suitably non-dimensionalized version of the wavefunction). The logarithmic Schrödinger equation is given by(5)(Cazenave 1983; Zwillinger 1997, p. 134), the nonlinear Schrödinger equation by(6)(Calogero and Degasperis 1982, p. 56;..

The system of partial differential equations(1)(2)

Partial differential equation boundary conditions which, for an elliptic partial differential equation in a region , specify that the sum of and the normal derivative of at all points of the boundary of , and being prescribed.

For the Helmholtz differential equation to be separable in a coordinate system, the scale factors in the Laplacian(1)and the functions and defined by(2)must be of the form of a Stäckeldeterminant(3)

The solutionwhere is the Riemann function of the linear Goursat problem with characteristics according to the Riemann method.

The system of partial differential equations(1)(2)

A partial differential equation ofsecond-order, i.e., one of the form(1)is called hyperbolic if the matrix(2)satisfies det. The wave equation is an example of a hyperbolic partial differential equation. Initial-boundary conditions are used to give(3)(4)(5)where(6)holds in .

The partial differential equation(Benton and Platzman 1972; Zwillinger 1995, p. 417; Zwillinger 1997, p. 130). The so-called nonplanar Burgers equation is given by(Sachdev and Nair 1987; Zwillinger 1997, p. 131).

The linear Boussinesq equation is the partialdifferential equation(1)(Whitham 1974, p. 9; Zwillinger 1997, p. 129). The nonlinear Boussinesq equation is(2)(Calogero and Degasperis 1982; Zwillinger 1997, p. 130). The modified Boussinesq equation is(3)(Clarkson 1986; Zwillinger 1997, p. 132).

There are three types of boundary conditions commonly encountered in the solution of partial differential equations: 1. Dirichlet boundary conditions specify the value of the function on a surface . 2. Neumann boundary conditions specifythe normal derivative of the function on a surface,3. Robin boundary conditions. For an elliptic partial differential equation in a region , Robin boundary conditions specify the sum of and the normal derivative of at all points of the boundary of , with and being prescribed.

A second-order partial differential equationarising in physics,If , it reduces to Laplace's equation.It is also related to the Helmholtzdifferential equation

On the surface of a sphere, attempt separation of variables in spherical coordinates by writing(1)then the Helmholtz differential equationbecomes(2)Dividing both sides by ,(3)which can now be separated by writing(4)The solution to this equation must be periodic, so must be an integer. The solution may then be defined either as a complex function(5)for , ..., , or as a sum of real sine and cosine functions(6)for , ..., . Plugging (4) into (3) gives(7)(8)which is the Legendre differential equation for with(9)giving(10)(11)Solutions are therefore Legendre polynomials with a complex index. The general complex solution is then(12)and the general real solution is(13)Note that these solutions depend on only a single variable . However, on the surface of a sphere, it is usual to express solutions in terms of the spherical harmonics derived for the three-dimensional spherical case, which depend on the two variables and ...

The system of partial differential equations(1)(2)

In two-dimensional polar coordinates, the Helmholtz differential equation is(1)Attempt separation of variables by writing(2)then the Helmholtz differential equationbecomes(3)Multiply both sides by to obtain(4)The solution to the second part of (4) must be periodic, so thedifferential equation is(5)which has solutions(6)Plug (5) back into (4)(7)This has solution(8)where and are Bessel functions of the first and second kinds, respectively. Combining the solutions gives the general solution(9)

A partial differential equation (PDE) is an equation involving functions and their partial derivatives; for example, the wave equation(1)Some partial differential equations can be solved exactly in the Wolfram Language using DSolve[eqn, y, x1, x2], and numerically using NDSolve[eqns, y, x, xmin, xmax, t, tmin, tmax].In general, partial differential equations are much more difficult to solve analytically than are ordinary differential equations. They may sometimes be solved using a Bäcklund transformation, characteristics, Green's function, integral transform, Lax pair, separation of variables, or--when all else fails (which it frequently does)--numerical methods such as finite differences.Fortunately, partial differential equations of second-order are often amenable to analytical solution. Such PDEs are of the form(2)Linear second-order PDEs are then classified according to the properties of the matrix(3)as..

In parabolic cylindrical coordinates, the scale factors are , and the separation functions are , giving Stäckel determinant of . the Helmholtz differential equation is(1)attempt separation of variables by writing(2)then the Helmholtz differential equationbecomes(3)Divide by ,(4)Separating the part,(5)(6)(7)so(8)which has solution(9)and(10)This can be separated(11)(12)so(13)(14)These are the Weber differential equations, and the solutions are known as Parabolic Cylinder Functions.

For a measurable function , the Beltrami differential equation is given bywhere is a partial derivative and denotes the complex conjugate of .

A partial differential equation ofsecond-order, i.e., one of the form(1)is called parabolic if the matrix(2)satisfies . The heat conduction equation and other diffusion equations are examples. Initial-boundary conditions are used to give(3)(4)where(5)holds in .

The scale factors are , and the separation functions are , , , given a Stäckel determinant of . The Laplacian is(1)Attempt separation of variables by writing(2)then the Helmholtz differential equationbecomes(3)Now multiply through by ,(4)Separating the part gives(5)which has solution(6)Plugging (5) back into (4) and multiplying by gives(7)Rewriting,(8)This can be rearranged into two terms, each containing only or ,(9)and so can be separated by letting the first part equal and the second equal , giving(10)(11)

In elliptic cylindrical coordinates, the scale factors are , , and the separation functions are , giving a Stäckel determinant of . The Helmholtz differential equation is(1)Attempt separation of variables by writing(2)then the Helmholtz differential equationbecomes(3)Now divide by to give(4)Separating the part,(5)so(6)which has the solution(7)Rewriting (◇) gives(8)which can be separated into(9)(10)so(11)(12)Now use(13)(14)to obtain(15)(16)Regrouping gives(17)(18)Let and , then these become(19)(20)Here, (19) is the mathieu differential equation and (20) is the modified mathieu differential equation. These solutions are known as mathieu functions.

In conical coordinates, Laplace'sequation can be written(1)where(2)(3)(Byerly 1959). Letting(4)breaks (1) into the two equations,(5)(6)Solving these gives(7)(8)where are ellipsoidal harmonics of the first kind. The regular solution is therefore(9)However, because of the cylindrical symmetry, the solution is an th degree spherical harmonic.

An ordinary differential equation (frequently called an "ODE," "diff eq," or "diffy Q") is an equality involving a function and its derivatives. An ODE of order is an equation of the form(1)where is a function of , is the first derivative with respect to , and is the th derivative with respect to .Nonhomogeneous ordinary differential equations can be solved if the general solution to the homogenous version is known, in which case the undetermined coefficients method or variation of parameters can be used to find the particular solution.Many ordinary differential equations can be solved exactly in the Wolfram Language using DSolve[eqn, y, x], and numerically using NDSolve[eqn, y, x, xmin, xmax].An ODE of order is said to be linear if it is of the form(2)A linear ODE where is said to be homogeneous. Confusingly, an ODE of the form(3)is also sometimes called "homogeneous."In general, an th-order ODE..

The ordinary Onsager equation is the sixth-order ordinarydifferential equation(Vicelli 1983; Zwillinger 1997, p. 128), while the partial Onsager equation is given by the partial differential equation(Wood and Martin 1980; Zwillinger 1997, p. 129).

A system of linear differential equations(1)with an analytic matrix, for which the matrix is analytic in and has a pole of order 1 at for , ..., . A system is Fuchsian iff there exist matrices , ..., with entries in such that(2)(3)

(1)Let , where denotes a Whittaker function. Then (1) becomes (2)Rearranging,(3)(4)so(5)where (Abramowitz and Stegun 1972, p. 505; Zwillinger 1997, p. 128). The solutions are known as Whittaker functions. Replacing by , the solutions can also be written in the form(6)where is a confluent hypergeometric function of the second kind and is a generalized Laguerre polynomial.

The twistor equation states thatwhere the parentheses denote symmetrization, in a Lorentz transformation, primed spinors transform under the conjugate of the transformation for unprimed ones, Einstein summation is used throughout, and denotes the spinor connection, which is equivalent to the Levi-Civita connection on Minkowski space. The zero rest mass equation can be solved by twistor functions. The solution uses ideas from complex variable theory and cohomology.

An indicial equation, also called a characteristic equation, is a recurrence equation obtained during application of the Frobenius method of solving a second-order ordinary differential equation. The indicial equation is obtained by noting that, by definition, the lowest order term (that corresponding to ) must have a coefficient of zero. 1. If the two roots are equal, only one solution can beobtained. 2. If the two roots differ by a noninteger, two solutionscan be obtained. 3. If the two roots differ by an integer,the larger will yield a solution. The smaller may or may not. For an example of the construction of an indicial equation, see Besselfunction of the first kind.The following table gives the indicial equations for some common differential equations.differential equationindicial equationBessel differential equationChebyshev differential equationHermite differential equationJacobi differential equationLaguerre differential..

A procedure for determining the behavior of an th order ordinary differential equation at a removable singularity without actually solving the equation. Consider(1)where is analytic in and rational in its other arguments. Proceed by making the substitution(2)with . For example, in the equation(3)making the substitution gives(4)The most singular terms (those with the most negative exponents) are called the "dominant balance terms," and must balance exponents and coefficients at the singularity. Here, the first two terms are dominant, so(5)(6)and the solution behaves as . The behavior in the neighborhood of the singularity is given by expansion in a Laurent series, in this case,(7)Plugging this series in yields(8)This gives recurrence relations, in this case with arbitrary, so the term is called the resonance or Kovalevskaya exponent. At the resonances, the coefficient will always be arbitrary. If no resonance term is..

Given an ordinary differential equation , the slope field for that differential equation is the vector field that takes a point to a unit vector with slope . The vectors in a slope field are usually drawn without arrowheads, indicating that they can be followed in either direction. Using a visualization of a slope field, it is easy to graphically trace out solution curves to initial value problems. For example, the illustration above shows the slope field for the equation together with solution curves for various initial values of .

There are a number of equations known as the Riccati differential equation. The most common is(1)(Abramowitz and Stegun 1972, p. 445; Zwillinger 1997, p. 126), which has solutions(2)where and are spherical Bessel functions of the first and second kinds.Another Riccati differential equation is(3)which is solvable by algebraic, exponential, and logarithmic functions only when , for , 1, 2, ....Yet another Riccati differential equation is(4)where (Boyce and DiPrima 1986, p. 87). The transformation(5)leads to the second-order linear homogeneous equation(6)If a particular solution to (4) is known, then a more general solution containing a single arbitrary constant can be obtained from(7)where is a solution to the first-order linear equation(8)(Boyce and DiPrima 1986, p. 87). This result is due to Euler in 1760...

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