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Solving a Second-Order Linear ODE with Periodic Forcing Using Fourier Series
To solve a linear differential equation driven by a repeating (periodic) input, break the input into a Fourier series, which is a sum of sines and cosines. Then solve the equation for one sine or cosine at a time, using the exponential response formula (a shortcut that divides by the characteristic polynomial evaluated at the input frequency). Finally add the pieces with the superposition principle, which says solutions to a linear equation can be summed. Use this when the forcing is a complicated periodic shape, such as a square wave, and you want a periodic particular solution.