= Exponentially conjugated wave equation
{title2=$(\partial_t+\gamma)^2u=c^2u_{xx},\quad v=e^{\gamma t}u$}
For real constants $\gamma,c$, the substitution $v=e^{\gamma t}u$ turns the displayed equation into the <wave equation> $v_{tt}=c^2v_{xx}$. If $u(x,0)=f(x)$ and $u_t(x,0)=g(x)$, then $v(x,0)=f(x)$ and $v_t(x,0)=g(x)+\gamma f(x)$. For $v_0=|c|>0$, <D'Alembert's formula> consequently gives
$$
\boxed{u(x,t)=e^{-\gamma t}\left[\frac{f(x-v_0t)+f(x+v_0t)}2+\frac1{2v_0}\int_{x-v_0t}^{x+v_0t}[g(s)+\gamma f(s)]\,ds\right].}
$$
The extra $\gamma f$ term is necessary even when the original initial velocity is zero. For $c=0$, take the limit to obtain $u=e^{-\gamma t}[f+t(g+\gamma f)]$. The formula gives classical solutions for suitably smooth data and <weak solutions> for less regular data. Positive $\gamma$ damps the transformed wave, whereas negative $\gamma$ amplifies it.
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