Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 5 / b

Write $q=i\omega$. A constant-coefficient linear ordinary differential equation has transfer function $1/P(q)$, where $P(q)={\sum }_{n=1}^c{a}_n{q}^n$. The specified simple poles correspond to $q=0,-2,1$. A minimal-order choice is thereforeThus one possible system isup to a common nonzero multiplicative constant. The root $q=1$ permits an exponentially growing homogeneous solution; the pole data therefore describe an unstable system. This does not prevent the requested algebraic reconstruction of its transfer function.