 Exponential substitution removing a constant first derivative

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.