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

Differential equation orders

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

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Differential equation orders

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- [Instructor] Let's dive deeper into some different characteristics of differential equations. This will help you classify the differential equations throughout this course. Let's start off by exploring the different orders of differential equations. The order of a differential equation is denoted by the derivative with the highest order present in a given equation. This tends to be one of the most common ways to classify a differential equation. So how do you determine the order of a differential equation? Let's begin with first-order differential equations. A first-order differential equation is when the highest derivative in the equation is a first derivative. This is denoted by dy/dx or y prime. Here are a few examples of first-order differential equations. For example, you might have y prime plus 2 multiplied by y equals x squared minus 4. Next, you might have dy/dx equals natural log of x, and finally, you might have y prime minus y equal to 0. Next, you have second-order differential equations. A second-order differential equation is when the highest derivative in the equation is a second derivative. This is denoted by d squared y over dx squared or y double prime. Let's look at a few examples. You might have y double prime minus y prime plus 2 multiplied by y equal to 0. Next is d squared y divided by dx squared minus y equals tangent of x. And finally, you have y double prime equal to e to the x minus 7. After second-order differential equations, you have what are often called higher-order differential equations. These can be third order, fourth order, and so on. These are denoted by d to the n of y of dx to the n or y to the n. You'll note that the n in this case is in parentheses to note that it is not the power of y, but it is simply denoting the derivative. Since sometimes the amount of ticks can get a lot to count once you get to these higher orders, let's look at a few examples of higher-order differential equations. First, you have y triple prime plus y prime minus 5 multiplied by y equals x squared minus 4. Next, you have g to the 4y of dx to the 4 minus d squared of y over dx squared equals 0. Finally, you have a fifth-order derivative of y to the 5th plus 8 multiplied by y double prime minus y prime is equal to e to the x plus x squared. The main thing to keep in mind is your order is determined by the highest order derivative in your equation. So even if you have multiple derivatives present in an equation, you're looking for the one with the highest order in order to classify it. For example, if you have the equation d squared y over dx squared plus dy/dx is equal to 15 multiplied by x plus 7. This would be a second-order differential equation because the highest order derivative is a second derivative, even though there is a first derivative present in the equation. Generally, first-order differential equations tend to be the easiest to work with. From there it is second order and so on where the complexity increases with the order. Generally, the techniques used for second-order differential equations are expanded upon for higher-order differential equations. It is important to note that the order does not depend on other ways to classify the equation, such as if it has ordinary or partial derivatives, or if it is linear or non-linear. These classifications will be explored in the following few videos. The first set of chapters in this course will focus on first-order differential equations. After that, you'll have a chapter on second order, and then a chapter on higher-order differential equations. You'll then expand onto other complex differential equation topics from there. As you have learned, you can simply identify the order of a differential equation by determining the highest derivative in your equation. Next, let's explore the differences between ordinary and partial differential equations.

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