Question

Solve the IVP (dirac's delta function)

Answer 1

\[y''+y=\delta(t-\pi)-\delta(t-2\pi)\]
\[y(0)=0\]
\[y'(0)=1\]

Answer 2

I got:
\[y=\frac{ e^{-\pi*s}-e^{-2*\pi*s}+1 }{ s^2+1 } = \sin(t) u(t-\pi)-\sin(t) u(t-2*\pi)+\sin(t)\]

The book's answer is: \[\sin(t) (0<t<\pi),\]\[0 (\pi<t<2\pi)\]\[-\sin(t) (t>2\pi)\]

Is my answer correct, and how would I get it in the above form?