Example 1: Solution to Differential Equations

Verify that the function $y=e^{–4x}+4x+8$ is a solution to the differential equation $y^{\prime }+4y=16x+36$.

Identifier: You are being asked to “verify that a function” is a “solution to the differential equation.

Step 1: Find the derivative of your given equation.

This derivative requires you to apply the Exponential Special Case, the power rule, and the derivative of a constant.

$y=e^{–4x}+4x+8$

$y^{\prime }=–4e^{–4x}+4$

Step 2: Compare your derivative from Step 1 to the differential equation you were given by the problem.

This example provided you a differential equation that includes the derivative of the original equation, $y^{\prime }$, and the original equation, $y$.

In order to see if what you have is a solution you will need to plug both the derivative you got out of Step 1, and the original equation you were given to start, into the differential equation you were given.

From there you need to simplify to see if the left side of the equals is the same as the right side of the equals.

From Step 1:   $y^{\prime }=–4e^{–4x}+4$

Given Differential Equation:    $y^{\prime }+4y=16x+36$

$y^{\prime }+4y=16x+36$

$\left((-4e^{–4x}+\mathrm{4)}\right)+4{\mathrm{(e}}^{–4x}+4x+\mathrm{8)}=16x+36$

$–4e^{–4x}+4+4e^{–4x}+16x+32=16x+36$

$16x+36=16x+36$

$16x+36=16x+36$

Final Result:

After finding your derivative and then plugging it and the original equation into the given differential equation, you were able to simplify the left side of the equals to show that it is identical to the right side of the equals. By doing this you have now verified that the function $y=e^{–4x}+4x+8$ is a solution to the differential equation$y^{\prime }+4y=16x+36$.