Pointers and help - solving an equation

I want to solve a differential equation, the exercise being, find two different independent solutions. (I don't know the proper English terms for this.)

I start out with . Differentiate to match each term in the equation and shove it in.

Recursive series.

Now I'm starting to get wobbly legs. I set to see if I can pinpoint .

results in a series . Can I render a solution from that even if it diverges?

results in a series that would imply that one solution is w = z. Or..?

Now I've sort of played all my cards here. What if I state a solution w = zu and solve the equation..? Or is there a simpler way to render an other solution here?