Exponential Distribution with changing (time-varying) rate parameter
Solution 1:
If $\lambda(t)=\mathrm e^t$, the density $f$ of the time until failure is such that, for every $t\geqslant0$, $$ f(t)=\mathrm e^{t+1-\mathrm e^t}. $$ More generally, $$ f(t)=\lambda(t)\,\mathrm e^{-\Lambda(t)},\quad\Lambda(t)=\int_0^t\lambda(s)\,\mathrm ds. $$ Note that every distribution with PDF $f$ and CDF $F$ can be written as such, using $$ \lambda(t)=\left\{\begin{array}{ccc}\frac{f(t)}{1-F(t)}&\mathrm{if}&F(t)\lt1,\\0&\mathrm{if}&F(t)=1.\end{array}\right. $$