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

Ordinary vs. partial differential equations

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

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Ordinary vs. partial differential equations

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- [Instructor] Now you know how to classify differential equations by their order. In this video, you will learn another way to classify differential equations. This time you will learn how to classify whether a differential equation is ordinary or partial. Ordinary differential equations contain only ordinary derivatives in them. These equations are sometimes abbreviated as ODE to stand for ordinary differential equation. In order for a derivative to be ordinary, this means that the derivative needs to be with respect to one variable. So for a differential equation, this means that all the derivatives are with respect to one variable, they are denoted by D to the N of Y over DX to the N or by the other ways I showed you prior on how to denote a derivative. For example, you may have D squared Y over DX squared minus three multiplied by DY DX plus Y equal to sign of x plus x squared. This would be an ordinary differential equation. Since there are all ordinary derivatives in it. Partial differential equations contain at least one partial derivative in them. These are sometimes abbreviated as PDE to represent partial differential equation. In this case, you need to have at least one derivative in your equation be with respect to two or more independent variables. This is denoted by a curly D to the N of Y over curly DX to the N. This curly D is actually a partial symbol, so it represents a partial derivative, so you'll just have to keep an eye out for if you have the regular D, it's a regular derivative or an ordinary derivative. If it is a curly D, then it is a partial derivative. So in this example, you may have a partial derivative of D squared Y over DX squared plus two, multiplied by the partial derivative of dy dx plus Y equals E to the X minus four. This would be a partial differential equation because there are two partial derivatives in it. Note that you can have a partial differential equation that contains one or more ordinary derivatives in it. For example, if you had the partial derivative of D squared Y over DX squared plus DY DX minus five, multiplied by y equals X cube minus two multiplied by x plus one. This contains both a partial derivative and an ordinary derivative. Note that if a differential equation though has at least one partial derivative, then it is automatically classified as a partial differential equation. So the equation you just saw, even though it contains an ordinary derivative in it, would still be a partial differential equation because it contains at least one partial derivative. If all the derivatives are ordinary in your equation, then it will be classified as an ordinary differential equation. So how can you tell if a differential equation is ordinary or partial? Usually you just need to look for an equation that contains the partial differentiation notation I showed you prior. Another good indication is if there are multiple variables being derived. Be careful though, since sometimes you can have an ordinary differential equation that contains multiple variables. Just note that the differential equation needs to be with respect to only one independent variable in order for it to be ordinary. Ordinary differential equations tend to be frequently used in systems such as electrical circuits and population dynamics. Partial differential equations tend to be used in physics equations such as wave heat and fluid dynamics. As you may be able to tell, ordinary differential equations tend to be much easier to work with due to only dealing with one independent variable. For this course, you'll mainly focus on understanding and solving ordinary differential equations. There will be one chapter near the end of the course though that will focus on understanding and solving partial differential equations. Now you know what ordinary and partial differential equations are. In this course, I will make sure to specify if a differential equation is partial, but otherwise it will be implied that the differential equations you work with are all ordinary. Up next, you'll learn about linear versus non-linear differential equations.

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