The first step to solving this equation was to use integration by parts, I let y1 = x^4 and y2 = ux^4. Then I would find the first and second derivatives of y2. After finding those I would plug it back into the equation and simplify. While doing this I would also need to use integration by parts again. Next using integrating factors to further get closer to the solution I will find that w(x) = a/x, and then I will integrate that to get the final solution that I would then plug back into the homogeneous general solution.
This is a question i struggle on during the exam. Seeing what you did really cleared up all the questions I had. I see I started right, but some where in my math I had went wrong. I am glad I saw how the questions is suppose to be done.AT least I know my story was right.