Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 15 / b

The first equation gives $v=u^{\prime }+3u$. Differentiate it and eliminate $v^{\prime }$ using the second equation:Thus the required second-order linear differential equation isConversely, a solution of this equation together with $v=u^{\prime }+3u$ solves both original equations, so the elimination loses no solutions.

The characteristic equation is $m^2+2m+2=0$, with roots $-1\pm i$. ThereforeIf the second component is also wanted, the relation $v=u^{\prime }+3u$ givesThese are damped oscillations associated with the complex conjugate eigenvalues of the coefficient matrix.

The first initial condition gives $A=1$. The original first equation gives $u^{\prime }(0)=-3u(0)+v(0)=-2$, whereas the general solution gives $u^{\prime }(0)=B-A$. Thus $B=-1$ andThe corresponding $v=e^{-t}(\cos t-3\sin t)$ has $v(0)=1$, checking the other initial condition.