For the verification part of the question, use the change of variable theorem (see for example Walpole, Myers and Myers (1998): Probability and Statistics for Engineers and Scientists, p 185).
If you need more help, please show all your work and say where you get stuck.
Example 7.4 seems to give the answer I am looking for.
My questions asks to verify by using U = X+Y and V=X-Y
Example 7.4 gives the distribution function for U. I think I have worked out the distribution function for V also as
f(v) = lamda/2 * (exp(-lamda *v) - exp(-3*lamda*v))
Is it now a question of working out f(x) based on X=U+V?