Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 2 / 8A / Solution

Termwise differentiation of the matrix exponential givesThus $y(t)=e^{tA}{y}_0$ satisfies $y^{\prime }=Ay$ and $y(0)=y_0$. If two solutions existed, multiplying their difference by $e^{-tA}$ would give a vector with zero derivative and zero initial value, proving uniqueness.

For the inhomogeneous equation, multiply by the integrating factor $e^{-tA}$:Integration and multiplication by $e^{tA}$ give the variation-of-constants formula

For the stated matrix,soWriting $f(t)=v\sin t$ with $v=(1,3,0{)}^T$, one findsThereforeor explicitly