Problem Description:

Solve the following 2^nd order equation using the method of undetermined coefficients:

Problem Solution:

1. Let’s first find the homogeneous solution. The characteristic equation is

with roots

Therefore the homogeneous solution is

where the last term is multiplied by x because cos 2x already appears within the homogeneous solution. Taking the first and second derivatives of this expression gives

or

Substitution of these expressions into the original linear ODE gives

First note that the terms containing and vanish. Now equating coefficients of like terms for the polynomial terms gives

Thus the first three coefficients become

Similarly for the sinusoidal terms, we have

or

Therefore, the particular solution is

3. Finally, combining the homogeneous and particular solutions gives the desired general solution, or

Example 2.2 explores a different approach to this equation using the Variation of Parameters method.