Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 2 / 7A / b / Solution

For $t<0$ both inputs vanish. Eliminating $y$ gives $x^{\prime \prime }+4x=0$, and the data at $t=-\pi$ giveFor all $t$, differentiating the first equation and using the second givesThus $x$ is continuous at zero while $x^{\prime }$ jumps by one. For $t>0$ the resulting solution isSince $\cos 2t=1-2{\sin }^{2}t$, this iswhereThus $a=1/2$, $b=1$, and the sketch of $q$ is a downward unit step at $t=\pi$.