To give an answer to this question, you should get a better idea of a differential equation. Basically, it's a math equation related to a specific function with derivatives. When it comes to its common applications, keep in mind that functions often represent different physical quantities, while derivatives represent the rate of change. This means that an equation determines a relationship that exists between them. As a math student, you should understand that such relations are very common, and that's why differential equations play an important role in a number of modern fields, including economics, engineering, biology, physics, and so on.
In mathematics, they are studied from a variety of perspectives related to their effective solutions or sets of functions. It's possible to use explicit formulas to solve only simple different equations. This means that solving a second order differential equation requires other methods. You can determine some basic properties of their solutions even without evaluating their exact form. If you don't have any self-contained formula for their effective solution, it can be numerically approximated with the help of computers. There are many numerical methods developed to find solutions with a certain degree of accuracy. Besides, theories of dynamical systems focus on their qualitative analysis described through differential equations.
As an example, when studying classical mechanics, you can easily describe the motion of any body by its particular velocity and position because its time value differs. Don't hesitate to dynamically express a set of important variables, including acceleration, as a certain differential equation for any unknown position of this body. Sometimes, it's possible to solve it explicitly.
There are some other aspects that you should learn before switching to a second order differential equation. Take a look at the example of modeling real-world problems through differential equations because it's all about determining the velocity of balls falling through the air. This is when only air resistance and gravity are considered. Don't forget that gravity is constant, but air resistance can be modeled proportionally to a velocity, which means that acceleration depends on the latter one. Finding it as time functions involves solving differential equations and verifying their validity.
They can be grouped into a few core types, and they can help choose the right approach to solving them. Make sure that you get the right knowledge of their important properties too.
Ordinary differential equations contain the function of 1 independent variable and derivatives. This term is used in contrast with partial differential equations that may include more than only 1 variable.
Linear differential equations have those solutions that are easy to add and multiply by coefficients. Besides, they are easy to understand, well-defined, and allow you to get closed-form solutions. When it comes to ordinary differential equations, they are nonlinear, and this means that their solutions are more intricate because you can't represent them by any elementary function in a closed form. That's because their solutions are analytic, exact, and come as integral forms or series. Take into account numerical and graphical methods that can be applied both by a computer and manually. They can help you get the right solutions for ordinary differential equations.
Before you get the definition of a second order differential equation, learn more about its partial type. Keep in mind that partial differential equations contain unknown multivariable functions and partial derivatives. They are often used to formulate certain problems that involve the functions of a few variables. You can solve them either manually or create a detailed computer model. It's interesting that such differential equations are also used to describe a number of phenomena, including heat, fluid flow, sound, elasticity, and others. Another important detail is that partial differential equations can model multidimensional systems.
Linear different equations include unknown functions and derivatives with a degree 1. Their main characteristic is that their available solutions form the subspace of the right function space, thus resulting in a more developed theory.
When dealing with homogeneous differential equations, remember that they are a subclass of the linear ones. They are all about the space of existing solutions in a linear subspace or a sum of all solutions. As a math student, you should know that the coefficients of unknown functions and derivatives in linear equations can be the functions of independent variables. If they are constant, then you're dealing with constant coefficient equation types.
Non-linear differential equations are easily formed by the products of unknown functions and derivatives with a degree higher than 1. It's another subject that should be studied before learning more about a second order differential equation. Unfortunately, there are only a few available methods to solve them exactly, and they are dependent on the equations that have particular symmetries. Take into account that such differential equations may have a complex behavior. It's necessary to formulate them correctly to get the right and easy solution.
Linear differential equations often appear as certain approximations to non-linear types, and they are valid only under a set of limited conditions. For instance, the harmonic oscillator type is the approximation to a pendulum equation, which is valid only for oscillations with small amplitude.
Finally, you also need to find out more about the right equation order because all differential equations are described by it. The good news is that their order is easy to determine the term with the highest number of derivatives. When solving the equations that contain only 1 derivative, they belong to the first-order type, while the one with double derivatives is called a second order differential equation.
There are many math students who agree that solving them is a tricky task. It's advisable to take a few basic steps to complete this academic process successfully. If you're not familiar with them, you can always count on the quality services provided by expert and reliable freelancers. They are easy to find over the Internet, and you only need to make the right choice, so be sure to compare available offers and samples. There are many benefits of getting their professional help, such as saving a lot of time to spend it on other important or urgent academic assignments. Another reason why they are becoming more and more popular is that their services are quite affordable, which makes them accessible to most students nowadays. Feel free to contact credible freelancers online if you have any problems with finding solutions for differential equations or other math subjects.