Start with the equation and we know a particular solution to the homogeneous equation is . Now let and and run through that routine in the book to get down to the expression . Integrate it once to get . Integrate it again to get . Back-substitute to obtain the solution
to the non-homogeneous equation.
Now the part with c1 and c2 represent the solution to the homogeneous case so that a particular solution is just the other part which I can write is as:
Now, if I expand into it's real and imaginary parts, then the real part, when substituted into the ODE, will give and the imaginary part will give and therefore, the function is a solution to: