Differential Equations
# Introduction

One of the challenges for new students of differential equations is the vast variety of equations they encounter. \(y \cdot y' \cdot y'' = x^2,\) \(\frac{1}{2x + 2y + y^{(10)}} = \sin(xy),\) \(\frac{\partial y}{\partial x_1} + 3x_1^2 \frac{\partial ^2 y}{\partial x_2^2} = x, …\) are all examples of differential equations.

Note, the notation \(\frac{\partial}{\partial x}\) refers to the partial derivative with respect to \(x\).

In this quiz, we try to bring some order to the chaos. We will classify various equations, and point out which ones we will learn to solve in this course.

Notational note: For higher-order derivatives, we often put the number of times you differentiate in parentheses. So \(y^{(10)}\) means the 10\(^\text{th}\) derivative of \(y.\) Without parentheses, though, \(y^{10}\) means you raise \(y\) to the 10\(^\text{th}\) power!

For starters, here is a simple distinction: If an equation only involves derivatives of a function of a single variable, it is called an **ordinary differential equation**. If it involves partial derivatives of a function of more than one variable, it is called a **partial differential equation**.

Partial differential equations are generally much harder to solve than ordinary differential equations. We'll touch on solving partial equations later, in chapter 5.

And here is a second classification: the **order** of a differential equation is the order of the highest derivative that appears.

What are the orders of the three examples from the previous question: \(y \cdot y' \cdot y'' = x^2,\) \(\frac{1}{2x + 2y + y^{(10)}} = \sin(xy'),\) \(\frac{\partial y}{\partial x_1} + 3x_1^2 \frac{\partial ^2 y}{\partial x_2^2} = x?\)

Here is another classification. An \(n^\text{th}\) order ordinary differential equation is **linear** if it can be written in the form
\[ p_n(x)\frac{d^ny}{dx^n} + p_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}} + \cdots + p_1(x)\frac{dy}{dx} + p_0(x)y = q(x).\]
This looks complicated, but all it's saying is that on the left we have a *linear combination* of the derivatives, where the coefficients are allowed to be functions of \(x.\) \((\)Note that in the last term, \(p_0(x)y,\) we think of the original function \(y\) as being like the "zeroth derivative".\()\) And on the right, we don't have any \(y\)'s at all.

The coefficients, as well as that \(q(x)\) on the right side, are allowed to be *any functions of \(x\)*, so linear differential equations may contain some very nonlinear looking expressions, like trig and exponentials. But the relationships between the *derivatives of \(y\)* are linear.

The following equations are all linear! \[\begin{align} y'' - 7y &= 0\\ x^2y' + \sin(x)y &= e^{\sin(x^2)}\\ x^3 &= \frac{1}{y + y''} \end{align}\] (For the last equation, clear out the denominator.)

An example of a nonlinear equation is \[y \cdot y' + 1 = x.\]

Our last classification is homogeneous vs. nonhomogeneous: A linear differential equation is **homogeneous** if the \(q(x)\) term on the right side of the expression from the last question is 0. In other words, homogeneous means there is no term that involves only \(x.\)

So \(x^3y'' + y' + \sin(x)y = 0\) is homogeneous, but \(x^3y'' + y' + \sin(x)y = x^2\) is nonhomogeneous.

How many of the following equations are second-order linear nonhomogeneous equations? \[\begin{align} y'' - 2y &= \sin(x)\\ e^x y'' - 2y &= \sin(x)\\ y'' - 3y &= \sin(y)\\ y'' - 4y - x &= 0\\ y'' - 5y - 17 &= 0 \end{align}\]

Let's get some perspective, and map out the rest of this Exploration!

Most differential equations are too hard for anyone to solve. We'll only investigate solving certain kinds of equations in this Exploration: mostly, we'll focus on linear equations. And when we do that, we'll look at homogeneous equations and nonhomogeneous separately.

In Chapter 2, we'll look at first-order equations. We'll see how to solve *all* linear first-order equations, plus a number of nonlinear ones too.
In Chapter 3 we turn to second-order. Unfortunately, we can't even solve all the linear ones here; we'll mostly focus on linear equations with *constant coefficients*.

The following flowchart summarizes the layout of Chapter 2 and 3:

Finally, in Chapter 4 we'll look at *systems* of differential equations (i.e. more than one equation at a time), and in Chapter 5 we'll introduce some additional topics, like partial differential equations.

According to the flowchart, where will we learn to solve \[y' = e^x \; y + x ?\]

According to the flowchart, where will we learn to solve \[y'' + 17y' - 4y = 0 ?\]

According to the flowchart, where will we learn to solve \[y'' + 3y' - 4y = \frac{x}{y} ?\]

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