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

Go to this website to view demonstrations of differential equations.

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

$\frac{d}{dx} \left( e^x \right) = e^x$
$\frac{d}{dx} \left( e^{g(x)} \right) = e^{g(x)}g'(x)$ (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 ${y}^{\prime }+3y=6x+11$ .

Show Solution

To verify the solution, we first calculate ${y}^{\prime }$ using the chain rule for derivatives. This gives ${y}^{\prime }=-3{e}^{-3x}+2$ . Next we substitute $y$ and ${y}^{\prime }$ into the left-hand side of the differential equation:

(- 3 e^{- 2 x} + 2) + 3 (e^{- 2 x} + 2 x + 3)

$\left(-3{e}^{-2x}+2\right)+3\left({e}^{-2x}+2x+3\right)$ .

The resulting expression can be simplified by first distributing to eliminate the parentheses, giving

- 3 e^{- 2 x} + 2 + 3 e^{- 2 x} + 6 x + 9

$-3{e}^{-2x}+2+3{e}^{-2x}+6x+9$ .

Combining like terms leads to the expression $6x+11$ , which is equal to the right-hand side of the differential equation. This result verifies that $y={e}^{-3x}+2x+3$ is a solution of the differential equation.

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 ${y}^{\prime }-3y=6x+4$ .

Hint

First calculate ${y}^{\prime }$ then substitute both ${y}^{\prime }$ and $y$ into the left-hand side.

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?

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

Show Solution

The highest derivative in the equation is ${y}^{\prime }$ , so the order is $1$ .
The highest derivative in the equation is $y\text{'''}$ , so the order is $3$ .
The highest derivative in the equation is ${y}^{\left(4\right)}$ , so the order is $4$ .

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?

(x^{4} - 3 x) y^{(5)} - (3 x^{2} + 1) y^{'} + 3 y = \sin x \cos x

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

Hint

Show Solution

$5$

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.)

A graph of a family of solutions to the differential equation y’ = 2 x, which are of the form y = x ^ 2 + C. Parabolas are drawn for values of C: -4, -2, 0, 2, and 4.

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 $\left(2,7\right)$ .

Show Solution

Any function of the form $y={x}^{2}+C$ is a solution to this differential equation. To determine the value of $C$ , we substitute the values $x=2$ and $y=7$ into this equation and solve for $C\text{:}$

\begin{matrix} y = x^{2} + C \\ 7 = 2^{2} + C = 4 + C \\ C = 3. \end{matrix}

$\begin{array}{}\\ \\ y={x}^{2}+C\hfill \\ 7={2}^{2}+C=4+C\hfill \\ C=3.\hfill \end{array}$

Therefore the particular solution passing through the point $\left(2,7\right)$ is $y={x}^{2}+3$ .

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^{'} = 4 x + 3

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

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

Hint

First substitute $x=1$ and $y=7$ into the equation, then solve for $C$ .

Show Solution

$y=2{x}^{2}+3x+2$

Module 4: Differential Equations