Let’s solve the differential equation .

First, we need to find the complementary solution by solving the associated homogeneous equation:

The characteristic equation is:

Thus, the complementary solution is:

where and are constants determined by initial conditions.

Next, we find a particular solution for the non-homogeneous part . We can use the method of undetermined coefficients.

Combining these, the trial solution is:

Next, we substitute into the original equation and equate coefficients for the different powers of , , and the trigonometric terms. This will give us a system of equations to solve for .

Once we compute those coefficients and solve for them, the general solution will then be:

where will include the specific polynomial, exponential, and harmonic components from our particular solution process.