Nonhomogeneous Linear ODE

Nonhomogeneous second-order ODEs model forced oscillators — and resonance is the famous edge case every quant interview loves.

Solve the initial value problem

$y^{\prime \prime }+y=\sin t,y(0)=0,y^{\prime }(0)=0.$

This is a forced harmonic oscillator driven exactly at its natural frequency. Explain why the standard particular-solution ansatz $y_p=A\sin t+B\cos t$ fails, and what to do instead.