= Exponential substitution removing a constant first derivative
{title2=$y=e^{ax}u:\quad y''-2ay'+(a^2+b^2)y=e^{ax}(u''+b^2u)$}
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.
Back to article page
