Differential Equations Workbook For Dummies®

7.2. Trying for a Solution in Polynomial Form

Whenever you come across a linear higher order differential equation that's nonhomo-geneous and in polynomial form, forget the other tricks to the method of undetermined coefficients and try for a polynomial of order n.

With that in mind, how would you handle this equation?

y″′ + 3y″ + 3y′ + y = x + 5

Obviously the general solution you need to find is the sum of the homogeneous solution and a particular solution. The formula for that looks like this:

y = y_h + y_p

The homogeneous version of the original differential equation is

y″′ + 3y″ + 3y′ + y = 0

Okay. Now what? Well, first you must solve this homogeneous equation and then plug in a particular solution of the form

y_p = Ax⁴ + Bx³ + Cx² + Dx + E

and solve for A, B, C, D, and E. Nothing to it, right? For the step-by-step process, take a look at the following example. Or if you think you have the hang of it from this general overview, skip ahead to the following two practice problems.

NOTE

EXAMPLE

Q. What's the solution to this differential equation?

y″′ + 3y″ + 3y′ + y = x + 5

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

1. First things first: Make sure you're looking for the sum of the particular solution and the solution to the homogeneous version of the differential equation:

   y = y_h + y_p

2. Now you can find the homogeneous version of the equation in the question:

   y″′ + 3y″ + 3y′ + y = 0

3. Because the homogeneous differential equation has constant coefficients, go ahead and assume a homogeneous solution ...