I have been stumped with this problem as well. I have tried several methods, but cannot seem to get the right answer.

I have two variables $X_1$ and $X_2$ which have the joint PDF $f(x_1,x_2) = e^{-x_1}e^{-x_2}$ if $x_1,x_2 > 0$ and $0$ otherwise. Now, we let $W = w_1X_1 + w_2X_2$ with $w_1,w_2 > 0$, and we need to confirm that the PDF is

$f(w) = \frac{1}{w_1-w_2} \left(e^{-\frac{w}{w_1}} - e^{-\frac{w}{w_2}}\right)$ when $w>0$ and $0$ otherwise.

Any help is appreciated! I can only seem to get one of the exponential terms and not both.