Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 2 / 7C (Differential Equations)

Spatial derivatives vanish for $u=y(t)$, giving $\ddot{y}+\gamma \dot{y}+{\omega }^{2}y=f(t)$.

The roots are $r=(-\gamma \pm \sqrt{{\gamma }^2-4{\omega }^2})/2$. They give oscillatory decay for $\gamma <2\omega$, two exponential decays for $\gamma >2\omega$, and $(A+Bt)e^{-\gamma t/2}$ at equality. All tend to zero.

Put $T=\pi /\omega$, $t_0=T/2$, and $\Omega =\omega /2$. A solution satisfying both endpoint conditions iswhere $A=-\sqrt{2}I{\omega }^{-1}{e}^{\sqrt{3}\pi /4}$. The derivative jumps by $I$ at $t_0$, as required.

Substitution gives $({\dot{X}}^2-c^2)\varphi \prime \prime -(\ddot{X}+\gamma \dot{X})\varphi \prime +{\omega }^2\varphi =0$. Multiply by $\varphi \prime$ and integrate over $\mathbb{R}$; decay kills the first and last integrals, leaving $(\ddot{X}+\gamma \dot{X})\int (\varphi \prime {)}^2=0$. Thus a nonconstant profile requires $X=X_{\infty }+C{e}^{-\gamma t}$ and approaches a fixed position.