Find a particular solution to the nonhomogeneous differential equation y'' + 4 y = cos(2x) + sin(2x).

The characteristic solution follows from solving the characteristic equation,



so that



A guess for the particular solution may be , but this is already contained within the characteristic solution. We require a set of linearly independent solutions, so we can look to



which has second derivative



Substituting into the ODE, you have





Therefore the particular solution is



Note that you could have made a more precise guess of



but, of course, any solution of the form is already accounted for within .