From the course: Complete Guide to Differential Equations Foundations for Data Science

Differential equation solutions

From the course: Complete Guide to Differential Equations Foundations for Data Science

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Differential equation solutions

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- [Presenter] Let's review the last main way to classify differential equations. Like with most equations, the goal is to find their solution. With differential equations, there are two main types of solutions you'll want to gather. These are general and particular. Let's first review the definition of a solution. A solution for a differential equation is any function that satisfies the differential equation on some open interval of A to B. The solutions are often accompanied with the corresponding interval. Note that you can have multiple possible solutions for a differential equation. Let's break this down into the two main types. First, a general solution represents the family of all possible solutions to a differential equation. This type of solution includes all the possible solutions due to the arbitrary constant value C it contains, hence the name "general solution." Let's look at a quick example. If you have a differential equation dydx equals three multiplied by x squared with the corresponding solution of y equal to x cubed plus C, then you know it is a general solution because it has that arbitrary constant C at the end. This means you could plug in any concept of value for C, making it have an infinite amount of solutions. A helpful aspect of the general solution of a differential equation is the number of constants it has corresponds to the order of the differential equation. So for example, if you have a second order differential equation, it would need two constant values. So if you see two constant values in a solution, then you could assume that it was for a second order differential equation. The example above would mean it is a first order differential equation because it only had one constant value. Let's move on to the other type of solution. A particular solution represents a specific solution to a differential equation. This means there are no arbitrary constants in the solution, meaning it is only one answer, hence the name "particular solution." These constants are solved by utilizing what are called initial conditions by plugging them into the given differential equation in order to get your one solution. Let's revisit the previous example. If you had your differential equation dydx equals three multiplied by x squared with an initial condition of y of zero equal to five, then you would have your general solution of y equals x cubed plus C, and then once you would solve with your initial condition, you would then get a particular solution of y equals x cubed plus five. Since it does not have any arbitrary constants left, it is now a particular solution. I want to note that there are also explicit and implicit solutions for differential equations. Explicit solutions are in the standard format of y equal to f of x. Any other format of a solution is known as an implicit solution. This course will focus on gathering explicit solutions since they tend to be the most logical ones to work with. Now you should be familiar with the differences between general and particular solutions for differential equations. Both types of solutions are often used in many industries such as physics, engineering, economics, and biology, depending on the needs of the equation being evaluated. Let's wrap up this chapter by exploring initial value problems.

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