Particular Solutions of Nonhomogeneous Differential Equations

In a Nut Shell: If the right hand side of the d.e. is a specified function, f(x), then the

second order, ordinary differential equation with constant coefficients is nonhomogeneous.

a d²y/dx² + b dy/dx + c y = f(x) (nonhomogeneous d.e.)

a d²y/dx² + b dy/dx + c y = 0 (homogeneous d.e.)

where a, b, and c are constants and f(x) is a specified function

Strategy: The first step in finding the particular solution is to set the right hand side of
the d.e. to zero and proceed to find the solution of the homogeneous equation. You are

then ready to move on to finding the particular solution.

Note: The functions appearing in the particular solution must differ (be linearly

independent) from those in the complementary solution.

More Background: There are two methods for finding particular solutions of

nonhomogeneous d.e.’s. The method of undetermined coefficients and the method of

variation of parameters. This section introduces the method of undetermined coefficients.

When can you use the method of undetermined coefficients? The answer is:

The method of undetermined coefficients can be used to find the particular solution

if the function, f(x), on the right hand side of the d.e. is one of the following types:

n A polynomial	n Sine functions	n Cosine functions
n Sine and Cosine functions	n Exponential functions	n Sums or products of these functions (Not quotients)

Click here to review the method to find complementary solutions of homogeneous d.e.’s.

Reminder: You need to first find the complementary solution to the homogeneous

equation so as not to repeat its functions in the particular solution,

If you are ready to find particular solutions of nonhomogeneous d.e.’s using the

method of undetermined coefficients, then click here for details and strategy.

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