From "Notes on Diffy Qs: Differential Equations for Engineers" Textbook by Jiri Lebl (Version 5.3):

Applications of Second Order Ordinary Differential Equations:

Derive a formula for x_sp if the equation is mx''+cx'+kx = F₀ cos(t)+F₁ cos(3t). Assume c > 0. (Remark: mathematically, x_sp is the particular solution of non homogeneous ordinary differential equation)

Please show each step of the derivation, thank you!