Solve (D*D + 4) y = x + cos²x

(D^2 + 9) y = cos 2x ... (1). The corresponding homogeneous equation is (D^2 + 9) y = 0, ... (2), whose auxiliary equation is m^2 + 9 = 0, which has (+/-) 3i as roots. The general solution of (2) is y = A. cos (3x) + B. sin (3x). Now to get a general solution of (1) we have just to add to the above, a particular solution of (1). One such solution is [cos (2x) ]/[-2^2 + 9] = (1/5). cos 2x. Hence a general solution of the given equation is given by y = A. cos (3x) + B. sin (3x) + (1/5) cos (2x), where A and B are arbitrary constants. The above solution incorporates all the solutions of the given equation.