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I'd like to give an alternative derivation not involving the clever (mystifying?) transformation to the heat equation and thus present a more general technique for solving constant coefficeint advection-diffusion PDEs. All we need is the Fourier transform: \begin{align*} \mathcal{F}[f] & = \int_{-\infty}^\infty e^{-i \omega y} f(y) dy, \end{align*} ...


For simplicity, we assume the necessary positivity, and then we can ignore the absolute signs. Note that \begin{align*} \big(C_1 - a e^{-bt} \big) d\omega = \big(C_1 - a e^{-bt} \big) f(t) dt + cbe^{-bt} \omega dt. \end{align*} That is, \begin{align*} \big(C_1 - a e^{-bt} \big)\,d\omega - cbe^{-bt} \omega \, dt= \big(C_1 - a e^{-bt} \big) f(t)dt. ...

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