y'' + y = e^x + x^3, y(0) = 2, y'(0)=0
There are two parts to this problem.
First we need to find the complimentery solution(the solution to the homogenious solution)
has the auxillary equation
so
Now we need to find the particular solution
Plugging these in we get
From this we see that both
and
So the solution by superposition
From here just plug in you I.C and you are done