## An Introduction to Differential Equations

**What is a Differential Equation?**

The study of differential equations can feel completely different than your previous mathematics courses. It is simultaneously the most methods-driven course of mathematics that you will have encountered thus far in your academic career, and the most theoretically ambitious course you will see in lower-division mathematics outside of linear algebra.

There is no illusion about the methodological complexity of this course. You will learn dozens of methods to solve differential equations, many of which rely on the equation being of a specific form or type. It can, and will, seem overwhelming at times; however, we will try our best to approach the topics in such a way as to make our decision-making process easier and more obvious. As much as I want to say, "Don't memorize. Understand!" The fact is that differential equations is a course where a good memory can pay massive dividends. Despite this, I will try my best to impart a level of understanding that will trump rote memorization.

Let's get started with our first definition.

There is no illusion about the methodological complexity of this course. You will learn dozens of methods to solve differential equations, many of which rely on the equation being of a specific form or type. It can, and will, seem overwhelming at times; however, we will try our best to approach the topics in such a way as to make our decision-making process easier and more obvious. As much as I want to say, "Don't memorize. Understand!" The fact is that differential equations is a course where a good memory can pay massive dividends. Despite this, I will try my best to impart a level of understanding that will trump rote memorization.

Let's get started with our first definition.

**DEFINITION | DIFFERENTIAL EQUATION**

An equation containing the derivative(s), either ordinary or partial, of one or more functions and/or dependent variables, in which the derivatives are with respect to one or more dependent variables, is called a

**differential equation (DE)**.

As an official definition, this is thorough, but

*almost*unnecessary. The reality is that a DE is*any*equation involving derivatives.
QUICK EXAMPLES\[y'' - y' + y = 1\] \[F(t,x,x') = m \frac{d^2x}{dt^2} \quad \text{(Newton's second law, } F=ma \text{)}\] \[x^3 \frac{d^3y}{dx^3} + 2x^2 \frac{d^2y}{dx^2} - x \frac{dy}{dx} + y = 12x^2\] \[x y'' = r \sqrt{1 + (y')^2}\] \[\frac{\partial^2 u}{\partial x \partial t} + \sin{(x)} \frac{\partial u}{\partial t} = t^2\] \[\tan{(y)} u_{xx} - \sin{(x)} u_{xy} + y u_{y} = 4\] |
Notice that I use several different notations here. It is assumed by this point that you are comfortable with the shorthand "prime" notation, as well as Leibniz notation. |

Since differential equations is a completely new subject for us, we are going to need a whole bunch of new words and phrases to help us describe the subject.

**Order of a Differential Equation**

In algebra, knowledge of the degree of a polynomial equation was often helpful for us to identify what method to use to solve that equation. If the equation was linear (degree 1), then we would collect all terms involving the variable on one side, if necessary, and then isolate that variable. If the equation was quadratic (degree 2), after some preliminary "clean up" we would choose from several solution methods based upon the form of the quadratic (i.e., extraction of roots, factoring, completing the square, or using the quadratic formula).

It turned out that first and second degree polynomials were the only ones with relatively simple methods of solutions. Methods for solving polynomial equations of degree greater than 2 often used techniques borrowed from solving second degree polynomial equations. In general, solving polynomial equations of degree greater than 4 is left for computers and numerical methods.

Along these same lines, we have a similar structure for differential equations. Instead of "degree," we talk about the order of the equation.

It turned out that first and second degree polynomials were the only ones with relatively simple methods of solutions. Methods for solving polynomial equations of degree greater than 2 often used techniques borrowed from solving second degree polynomial equations. In general, solving polynomial equations of degree greater than 4 is left for computers and numerical methods.

Along these same lines, we have a similar structure for differential equations. Instead of "degree," we talk about the order of the equation.

**DEFINITION | ORDER OF A DIFFERENTIAL EQUATION**

The

**order**of a differential equation is the order of the highest derivative in the equation.

**QUICK EXAMPLES**

\[y'' - y' + y = 1 \quad \text{second order}\]

\[x^3 \frac{d^3y}{dx^3} + 2x^2 \frac{d^2y}{dx^2} - x \frac{dy}{dx} + y = 12x^2 \quad \text{third order}\]

\[\frac{\partial^2 u}{\partial x \partial t} + \sin{(x)} \frac{\partial u}{\partial t} = t^2 \quad \text{second order}\]

\[\tan{(y)} u_{xx} - \sin{(x)} u_{xy} + y u_{y} = 4 \quad \text{second order}\]

The vast majority of differential equations courses focus only on first- and second-order differential equations. The reason for this limitation is pretty simple - just like solving polynomial equations gets too complex as the degree increases, solving higher-order differential equations analytically pretty much becomes impossible.

We will have less than two handfuls of methods for solving first-order, and a whole bunch of other methods for solving second-order differential equations. Some of the methods for solving second-order differential equations can be extended to solve higher-order equations, but there are going to be a lot of "if it looks like this, then do this."

Before we go any further, I should mention the two major classes of differential equations that can be encountered.

We will have less than two handfuls of methods for solving first-order, and a whole bunch of other methods for solving second-order differential equations. Some of the methods for solving second-order differential equations can be extended to solve higher-order equations, but there are going to be a lot of "if it looks like this, then do this."

Before we go any further, I should mention the two major classes of differential equations that can be encountered.

**Major Classes of Differential Equations**

**DEFINITION | ORDINARY DIFFERENTIAL EQUATION (ODE)**

A differential equation consisting only of ordinary (non-partial) derivatives is called an

**ordinary differential equation**(

**ODE**).

**QUICK EXAMPLES**

\[\frac{dy}{dx} + xy = 3\] \[y'' - y' + y = 1\] \[\frac{d^2x}{dt^2} = a \quad \text{(acceleration)}\] \[F(t,x,x') = m \frac{d^2x}{dt^2} \quad \text{(Newton's second law, } F=ma \text{)}\] \[x^3 \frac{d^3y}{dx^3} + 2x^2 \frac{d^2y}{dx^2} - x \frac{dy}{dx} + y = 12x^2\] \[x y'' = r \sqrt{1 + (y')^2}\]

**DEFINITION | PARTIAL DIFFERENTIAL EQUATION (PDE)**

A differential equation containing partial derivatives is called a

**partial differential equation**(

**PDE**).

**QUICK EXAMPLES**

\[\frac{\partial^2 u}{\partial x \partial t} + \sin{(x)} \frac{\partial u}{\partial t} = t^2\] \[\tan{(y)} u_{xx} - \sin{(x)} u_{xy} + y u_{y} = 4\]

The study of differential equations is broken up into these two classes - literally. This course is focused

Finally, it is necessary to introduce the common forms in which we write differential equations.

*almost*completely on solving ordinary differential equations. You will have the opportunity in upper division courses to learn how to conceptualize and solve partial differential equations.Finally, it is necessary to introduce the common forms in which we write differential equations.

**Forms for Ordinary Differential Equations**

**DEFINITION | GENERAL FORM**

An \(n^{th}\)-order ordinary differential equation is said to be in

**general form**when all non-zero terms are isolated on a common side of the equation as in\[F(x,y,y',y'',\ldots,y^{(n)}) = 0.\]

**DEFINITION | NORMAL FORM**

An \(n^{th}\)-order ordinary differential equation is said to be in

**normal form**if the \(n^{th}\) derivative is isolated on one side of the equation as in\[y^{(n)} = f(x,y,y',y'',\ldots,y^{(n-1)}).\]

Types of Differential Equations

**DEFINITION | AUTONOMOUS DIFFERENTIAL EQUATIONS**

A differential equation that does not explicitly contain the independent variable in any of its terms is called an

**autonomous differential equation**(or just

**autonomous**).

**DEFINITION | LINEAR DIFFERENTIAL EQUATION, SOURCE TERM**

An \(n^{th}\)-order differential equation is called

**linear**if it can be written in the form\[y^{(n)} = \sum_{i = 0}^{n - 1}{a_i(x) f^{(i)}(x)} + r(x),\]where \(a_i(x)\) and \(r(x)\) are continuous. In this case, \(r(x)\) is called the

**source term**.

DEFINITION | HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONA linear differential equation is called homogeneous if \(r(x) = 0\).DEFINITION | NONHOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONA linear differential equation is called nonhomogeneous if \(r(x) \neq 0\). |

**DEFINITION | NONLINEAR DIFFERENTIAL EQUATION**

A differential equation that is not linear is called

**nonlinear**.

**DEFINITION | SEPARABLE EQUATION**

A first-order differential equation of the form \[\frac{dy}{dx} = g(x) \cdot h(y)\]is said to be

**separable**.

Recognizing Familiar Patterns in Differential Equations