Past exam of the mathematical methods I course of the University of Oxford / 2018 / 9 / b / Solution

The linear system of differential equations has matrixThe columns of $C$ are eigenvectors of $M$. The change of basis $v^{\prime }=C^{-1}v$ therefore gives $v_1^{\prime }=(v_1-i{v}_2)/2$, $v_2^{\prime }=(v_1+i{v}_2)/2$, with independent first-order linear differential equationsThe initial velocity gives $K_{+}=-i{v}_0/2$, $K_{-}=i{v}_0/2$. Transforming back, $v=C{v}^{\prime }$, yields the real velocity vector componentsThese also cover $\alpha =0$. Complex coordinates have decoupled a real rotation without making the physical velocity complex.