Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 4 / 12B (Complex Methods)

Because the matrix-valued Laplace transform is taken componentwise, for every $i,j$ we haveThe sum is finite and $A$ is constant, so it may be moved outside the integral. Therefore

Taking the Laplace transform of the differential equation and usinggivesHenceWhenever $s$ is not an eigenvalue of $A$, the matrix $sI-A$ is invertible, and thusThis is the Laplace-transform solution of a constant-coefficient vector ODE.

Set $g=0$. The given homogeneous solution is $y(t)=e^{tA}{y}_0$, so the preceding formula saysfor every initial vector $y_0$. Equality on every vector gives the matrix identityon the common domain of convergence and in particular away from the eigenvalues of $A$. This is the Laplace transform of a matrix exponential.

The coefficient matrix has eigenvector $u_{+}=(1,1{)}^T$ with eigenvalue $3$ and eigenvector $u_{-}=(1,-1{)}^T$ with eigenvalue $-1$. WriteSinceand $y(0)=(1,-2{)}^T=-\frac{1}{2}{u}_{+}+\frac{3}{2}{u}_{-}$, the system diagonalises toUsing an integrating factor givesThe unique fastest term is thereforeConsequently $e^{-nt}y(t)$ diverges for $n<3$, tends to zero for $n>3$, and for $n=3$ tends to the finite nonzero vectorThus the only requested integer isThis is an instance of dominant eigenmode in a forced linear system.