Exponential substitution removing a constant first derivative ($y=e^{ax}u:y^{\prime \prime }-2a{y}^{\prime }+(a^2+b^2)y=e^{ax}(u^{\prime \prime }+b^{2}u)$)

Differentiating $y=e^{ax}u$ twice proves the displayed identity. It converts an exponentially weighted forced equation into a simple harmonic oscillator equation. If $b=0$, the reduced homogeneous solutions are $1,x$ rather than $\cos bx,\sin bx$ as an independent pair. If the forcing contains a term at frequency $|b|$, resonance requires an additional polynomial factor in a particular solution.