Past exam of the mathematics course of the University of Cambridge / 2016 / ia / Paper 2 / 8A / a / Solution

Write the system as ${\mathbf{x}}^{\prime }=M\mathbf{x}$, with $M=\left(\begin{array}{cc}2& 5\\ -1& -2\end{array}\right)$. Its characteristic polynomial is ${\lambda }^2+1$, so its eigenvalues are $i$ and $-i$. An eigenvector for $i$ is $v=(5,-2+i{)}^{\mathsf{T}}$. Thus $e^{it}v$ is a complex solution, and its real part and imaginary part give real solutions:Their initial vectors are $(5,-2{)}^{\mathsf{T}}$ and $(0,1{)}^{\mathsf{T}}$, with determinant $5\ne 0$. Consequently they are linearly independent and span the two-dimensional solution space.