Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2008 / 9 / iv / Solution

Applying the derivative rule twice and using both zero initial data givesso $\bar{x}(p)=1/(p^2+1{)}^2$. By the convolution theorem for Laplace transforms,Use the product-to-sum identity $\sin s\sin (t-s)=\frac{1}{2}[\cos (2s-t)-\cos t]$ to integrate:It satisfies $x(0)=x^{\prime }(0)=0$, and differentiation gives $x^{\prime \prime }+x=\sin t$. The term proportional to $t\cos t$ is the resonant response of the forced harmonic oscillator.