Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 2 / 15 / a

The characteristic equation has the repeated polynomial root $m=-2$, because both $e^{-2x}$ and $x{e}^{-2x}$ occur as independent solutions. Hence its characteristic polynomial isComparing constant terms gives

The characteristic equation factors asThe distinct characteristic roots therefore give the general solution

The characteristic roots areTaking real linear combinations of the two complex exponential solutions yieldsThe coefficient $\beta$ is three here, as confirmed in the PDF; the OCR replaces it by an undefined symbol.