It's a "Show That". All you need is a couple of derivatives and some algebra.
Let's see what you get for the 1st and 2nd derivative of the provided expression.
for part a) Find and and substitute it into the equation given you should find that the LHS=RHS which shows its a solution.
for the second part you need to solve the complimentary function
this would have an auxilary quadratic of
solving that you get as a repeated root so the solution to the complimentary function is
where A and B are the constants of integration.
Putting this together with the solution we were given in part a) we have
so all we have to do is find the constants from the given boundary conditions.
I get A=1 and B=1 do post again if you have any problem with doing that or understanding what I've written
Hope that helps
In the last line :
Also, be careful when using tex for multiplications, don't use x because it's just like the variable and we can easily get confused. You can use ( \cdot) or ( \times)
Plus, you made some typos like d/dx (uv)=(xe^x), which is false to write. I guess you wanted to say :
Want me to try showing you how to write all this stuff, or is it ok ?