Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 2 / 7A (Differential Equations)

For every test function $\phi$ and nonzero $p$, substitution $u=pt$ with the orientation handled by $|p|$ givesHence the Dirac delta scaling identity is $\delta (pt)=\delta (t)/|p|$.

The derivative of a distribution is defined by integration by parts. Therefore

For a compactly supported test function $f$,Thus this is the distributional derivative of the Heaviside step function: $H^{\prime }=\delta$.

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 $t=\pi$.