Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 2 / 12 / i / Solution

The characteristic equation is $r^2+\lambda r+k=0$, whose roots are $r_{\pm }=(-\lambda \pm \sqrt{{\lambda }^2-4k})/2$. For distinct real roots, $y=C_1{e}^{r_{+}t}+C_2{e}^{r_{-}t}$; at the repeated-root threshold ${\lambda }^2=4k$, $y=(C_1+C_{2}t)e^{-\lambda t/2}$. In the complex-root case put $\Omega =\sqrt{k-{\lambda }^2/4}$ to obtainThis is precisely the oscillatory regime of the damped harmonic oscillator. The zero solution is of course nonoscillatory; the regime describes its nontrivial motions.