Worked example - driven damped oscillator at resonance. x_ddot + 4 x = 8 cos(2 t) (m = 1, k = 4, no damping; natural frequency omega_0 = 2 = driving frequency: resonance!). Step 1: homogeneous: r^2 + 4 = 0 -> r = +-2i, so x_h = C cos 2t + D sin 2t. Step 2: guess x_p = A t cos 2t + B t sin 2t (multiply by t since cos 2t is in x_h). Compute: x_p' and x_p''; substituting gives -4A sin 2t + 4B cos 2t = 8 cos 2t (the t terms cancel - verify by expansion). So B = 2, A = 0: x_p = 2 t sin 2t. Step 3: full solution x = C cos 2t + D sin 2t + 2 t sin 2t - the amplitude grows without bound, pure resonance. With damping b > 0, the growth saturates at amplitude F_0/(b omega).