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

What are differential equations?

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

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What are differential equations?

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- [Instructor] In the previous video, you reviewed what derivatives are. Now you are ready to explore what differential equations are since they are highly related. Let's get started. A differential equation is an equation that contains at least one derivative of an unknown function. An easy way to tell if an equation is a differential equation is if it contains any derivatives in it. Now you see why derivatives are such a key part of differential equations is because they are what makes them unique. These are some examples of differential equations. First, you have y double prime minus two multiplied by y prime plus five multiplied by y equal to zero. Next, you have dy/dx plus y equals e to the x plus three, and finally, you have y triple prime minus eight multiplied by y prime equals sine of x plus cosine of x. As you may notice, there are different variations of how the derivative is formatted, but you just need to make sure that each of these equations has at least one derivative, which they all do. The main goal of differential equations is solving them for their original functions. This will get rid of any derivatives in the equation and provide you with the original function you can then use. As you may have noticed, the process of solving differential equations sounds a whole lot like integrating in calculus. Know that they are closely related, and you will use integration throughout this course. Differential equations can prove to be more complex, though, than a simple integration, hence, why it is its own separate topic in mathematics. Let's look at a physics example. Newton's second law of motion is an excellent example of a differential equation. Here it has the equation of F equal to m multiplied by a. Here, F is force, m is mass, and a is acceleration. When solving this equation, you will usually either solve it in respect to the first derivative of velocity or the second derivative of position. Hence, this makes it a differential equation. Throughout this chapter, you will explore various ways to classify differential equations. This includes the order of the differential equation, whether it's ordinary or partial, whether it's linear or non-linear, whether you're gathering a general or particular solution. And finally, you'll learn how to work with initial value problems. There are many real life applications for differential equations. In finance, differential equations are used to model and predict market trends. In machine learning, they're used to model complex systems and work with continuous dynamics, such as simulating physical phenomena. In physics, they're used with various equations, such as Newton's laws, wave propagation, and heat transfer. You can also use them in engineering for fluid dynamics, circuit analysis and control systems. You'll notice me mention a lot of these applications throughout this course. Now you're ready to explore differential equations. A big portion of this course will focus on understanding what differential equations are, how to solve them, and how to apply them. Then you'll move on to advanced differential equation topics such as series, systems, and Laplace transforms once you're comfortable solving differential equations. Before you dive into classifying differential equations, let's take a moment to explore the concept of direction fields.

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