How do you find the steady periodic solution of this ODE?

Question

Find the steady state periodic solution of the differential equation: $$ y^{''}+16y = f(t)$$

where $f(t)$ is an odd function of period $2\pi$ such that $$f(t) = t $$ $$ 0 \leq t < \pi $$

Answer 1

There is no periodic solution. All solutions of the differential equation have $y((2n+1)\pi) = y((2n-1)\pi) + \pi/8$.