## Solutions to Differential Equations

### Learning Outcomes

• Identify the order of a differential equation
• Explain what is meant by a solution to a differential equation
• Distinguish between the general solution and a particular solution of a differential equation

## General Differential Equations

Consider the equation ${y}^{\prime }=3{x}^{2}$, which is an example of a differential equation because it includes a derivative. There is a relationship between the variables $x$ and $y\text{:}y$ is an unknown function of $x$. Furthermore, the left-hand side of the equation is the derivative of $y$. Therefore we can interpret this equation as follows: Start with some function $y=f\left(x\right)$ and take its derivative. The answer must be equal to $3{x}^{2}$. What function has a derivative that is equal to $3{x}^{2}?$ One such function is $y={x}^{3}$, so this function is considered a solution to a differential equation.

### Definition

A differential equation is an equation involving an unknown function $y=f\left(x\right)$ and one or more of its derivatives. A solution to a differential equation is a function $y=f\left(x\right)$ that satisfies the differential equation when $f$ and its derivatives are substituted into the equation.

### Interactive

Some examples of differential equations and their solutions appear in the following table.

Equation Solution
$y^{\prime} =2x$ $y={x}^{2}$
$y^{\prime} +3y=6x+11$ $y={e}^{-3x}+2x+3$
$y^{\prime} \prime -3y^{\prime} +2y=24{e}^{-2x}$ $y=3{e}^{x}-4{e}^{2x}+2{e}^{-2x}$

Note that a solution to a differential equation is not necessarily unique, primarily because the derivative of a constant is zero. For example, $y={x}^{2}+4$ is also a solution to the first differential equation in the table. We will return to this idea a little bit later in this section. First, we briefly review the rules for derivatives of exponential functions, and then explore what it means for a function to be a solution to a differential equation.

### Recall: Derivatives of Exponential Functions

1. $\frac{d}{dx} \left( e^x \right) = e^x$
2. $\frac{d}{dx} \left( e^{g(x)} \right) = e^{g(x)}g'(x)$    (this is the chain rule applied to the derivative of an exponential function)

### Example: Verifying Solutions of Differential Equations

Verify that the function $y={e}^{-3x}+2x+3$ is a solution to the differential equation ${y}^{\prime }+3y=6x+11$.

Watch the following video to see the worked solution to Example: Verifying Solutions of Differential Equations

You can view the transcript for “4.1.2” here (opens in new window).

### try it

Verify that $y=2{e}^{3x}-2x - 2$ is a solution to the differential equation ${y}^{\prime }-3y=6x+4$.

It is convenient to define characteristics of differential equations that make it easier to talk about them and categorize them. The most basic characteristic of a differential equation is its order.

### Definition

The order of a differential equation is the highest order of any derivative of the unknown function that appears in the equation.

### Example: Identifying the Order of a Differential Equation

What is the order of each of the following differential equations?

1. ${y}^{\prime }-4y={x}^{2}-3x+4$
2. ${x}^{2}y\text{'''}-3xy\text{''}+x{y}^{\prime }-3y=\sin{x}$
3. $\frac{4}{x}{y}^{\left(4\right)}-\frac{6}{{x}^{2}}y\text{''}+\frac{12}{{x}^{4}}y={x}^{3}-3{x}^{2}+4x - 12$

Watch the following video to see the worked solution to Example: Identifying the Order of a Differential Equation

You can view the transcript for “4.1.1” here (opens in new window).

### try it

What is the order of the following differential equation?

$\left({x}^{4}-3x\right){y}^{\left(5\right)}-\left(3{x}^{2}+1\right){y}^{\prime }+3y=\sin{x}\cos{x}$

## General and Particular Solutions

We already noted that the differential equation ${y}^{\prime }=2x$ has at least two solutions: $y={x}^{2}$ and $y={x}^{2}+4$. The only difference between these two solutions is the last term, which is a constant. What if the last term is a different constant? Will this expression still be a solution to the differential equation? In fact, any function of the form $y={x}^{2}+C$, where $C$ represents any constant, is a solution as well. The reason is that the derivative of ${x}^{2}+C$ is $2x$, regardless of the value of $C$. It can be shown that any solution of this differential equation must be of the form $y={x}^{2}+C$. This is an example of a general solution to a differential equation. A graph of some of these solutions is given in Figure 1. (Note: in this graph we used even integer values for $C$ ranging between $-4$ and $4$. In fact, there is no restriction on the value of $C$; it can be an integer or not.)

In this example, we are free to choose any solution we wish; for example, $y={x}^{2}-3$ is a member of the family of solutions to this differential equation. This is called a particular solution to the differential equation. A particular solution can often be uniquely identified if we are given additional information about the problem.

### Example: Finding a Particular Solution

Find the particular solution to the differential equation ${y}^{\prime }=2x$ passing through the point $\left(2,7\right)$.

Watch the following video to see the worked solution to Example: Finding a Particular Solution

You can view the transcript for “4.1.3” here (opens in new window).

### try it

Find the particular solution to the differential equation

${y}^{\prime }=4x+3$

passing through the point $\left(1,7\right)$, given that $y=2{x}^{2}+3x+C$ is a general solution to the differential equation.