The Wronskian & Variation of Parameters · Medium problem

Solving $y''+y=\sin t$ by variation of parameters with $y_1=\cos t,\ y_2=\sin t$ gives $u_2'=\cos t\sin t=\tfrac12\sin2t$, so $u_2(t)=-\tfrac14\cos2t+C$. Given that $u_1(0)=0$, find the value of $C$ for which $y_p'(0)=0$.