Differential Equations Workbook For Dummies®

7.3. Working with Solutions Made Up of Sines and Cosines

Spotting a sine or cosine in a nonhomogeneous linear higher order differential equation you're facing is a surefire sign that you need to find a particular solution that includes sines and cosines so you can plug it into the equation and solve for A and B.

Say you're tackling this differential equation:

y″′ + 7y″ + 14y′ + 8y = 5 sin (x)

You're well aware that the general solution is of this form:

y = y_h + y_p

where y_h equals the homogeneous solution and y_p equals the particular solution.

The homogeneous version of your original equation is

y″′ + 7y″ + 14y′ + 8y = 0

What you need to do next is solve this homogeneous equation first and then plug in a particular solution of the form

y_p = A sin (x) + B cos (x)

so you can solve for A and B.

The following example shows you how to solve this type of equation. Spend a few minutes reviewing it before trying out a couple practice problems on your own.

NOTE

EXAMPLE

Q. Find the solution to this differential equation:

y″′ + 7y″ + 14y′ + 8y = 2 sin (x) + 26 cos (x)

A. y = c₁e^−x + c₂e^−2x + c₃e^−4x + 2 sin (x)

1. You already know that the general solution looks like this:

   y = y_h + y_p

   so start by finding the homogeneous version of the differential equation:

   y″′ + 7y″ + 14y′ + 8y = 0

2. You can assume a homogeneous solution of the following form because the homogeneous version of the equation has constant coefficients:

   y = e^rx

3. Plug in your attempted solution to get

   r³e^rx + 7r²e^rx + 14re^rx + 8e^rx = 0

4. Then ...