Differential Equations Workbook For Dummies®

7.1. Seeking Out Solutions of the Form Ae^rx

NOTE

Solving for a solution in the form of Ae^rx is what you should try first when you want to find a solution to the homogeneous version of a nonhomogeneous linear higher order equation, like this one:

y″′ + 3y″ + 3y′ + y = 432e^5x

You know that the general solution of this equation is the sum of the particular solution and the homogeneous solution. You also know that in variable-speak, the general solution looks like this:

y = y_h + y_p

Perfect. So now you need to get the homogeneous version of the previous differential equation, which is

y″′ + 3y″ + 3y′ + y = 0

Solve the homogeneous equation first; then plug in a particular solution of the form

y_p = Ae^5x

All that's left to do is solve for A!

That was just a quick outline of the process; the following example walks you through the steps of solving the previous equation from start to finish. Check it out and then take a shot at solving the related practice problems.

NOTE

EXAMPLE

Q. Find the solution to this differential equation:

y″′ + 3y″ + 3y′ + y = 432e^5x

A. y = c₁e^−x + c₂xe^−x + c₃x²e^−x + 2e^5x

1. Start by realizing that obtaining the general solution to the problem means finding the sum of the particular solution and the solution to the homo-geneous version of the differential equation:

   y = y_h + y_p

2. Then find the homogeneous version of the differential equation:

   y″′ + 3y″ + 3y′ + y = 0

3. The homogeneous version has constant coefficients, so you can assume a homogeneous solution of the form

   y