Variation of Parameters Problem
This is the first time I have posted here and my first time using latex. Hope it comes out OK!
I have a problem to solve for my diff eq's class. It regards using variation of parameters to solve inhomogeneous, second-order DE's. I've been beating my head against the wall with this one and am very frustrated. Hopefully someone can give me some direction.
The problem says to find a particular solution to the DE
where y1 and y2 are solutions to the corresponding homogeneous equation and g(x) is an arbitrary function of x.
Dividing through by I get
My first thought was to use the formula for finding a particular solution
Where the denominator in both integrands is the Wronskian of y1 and y2. I calculate the Wronskian for my problem to be
So now I try to set up the integrals to calculate a particular solution as
It is at this point that I am totally confused. I just don't know how to proceed with an arbitrary function of x in the integrand. To make matters worse, I peeked at the answer in the back of the book. It shows
How the answer reduces to a single integral with both x and t in the integrand has me completely flustered. It would seem that the fact x is between 0 and 1 would be a clue, but I just can't figure out how to use it.
I am not looking for a complete, worked solution. But if someone could point me in the right direction, I would really appreciate it.
Thanks in advance.