Hello!

I having problems getting started with this problem:

Using subn $\displaystyle t = e^x $ and $\displaystyle z(t) = y(x) $ rewrite:

$\displaystyle \frac{d^2y}{dx^2} - \frac {dy}{dx} + e^{2x}y = xe^2x -1 $

hence find all solutions.

When I perform the substitution using$\displaystyle t^2 = e^{2x} $ and $\displaystyle x = lnt$, I get:

$\displaystyle \frac{d^2z}{dt^2} - \frac{dz}{dt} + t^2z = t^2lnt +1$

I can't find any complimentary functions and probably due to my undergrad engineering major, I tried a Laplace transform, but end up with a horrible function owing to the $\displaystyle t^2z$ part that would leave a $\displaystyle \frac{d^2s}{dt^2}$ on the LHS- I suspect this is not be the correct method as alluded to within the question - there must be a reason for the subn.....

I can't use an order reduction or variation of parameters as I don't have an already known solution- unless I use a trial and error! If anyone can kick me in the right direction, I will be most greatful.... I can't seem to see through it right now!

Thanks for looking!