Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 2 / 7A / a / Solution

Write the coefficient matrix as $M$. Its characteristic polynomial is $\det (rI-M)=r(r-2)(r+2)$. Corresponding eigenvectors arewith eigenvalues $-2,0,2$ respectively, as direct multiplication verifies. Distinct eigenvalues make these vectors a basis. Expanding a solution as $z_{-}{v}_{-}+z_0{v}_0+z_{+}{v}_{+}$ decouples the linear ordinary differential equations into ${\dot{z}}_{-}=-2z_{-}$, ${\dot{z}}_0=0$, ${\dot{z}}_{+}=2z_{+}$. Thus the general solution is