The second one is Bessel's equation, I think. I'll Google for the solution.
If anyone has an idea for the other two..
Yes, I know, I edited the post above, I thought you were talking about the first one. It's not an excuse, but it's late in this timezone. :-)
Oh, thank you! And I will read about series solutions, maybe they aren't that hard to learn.i will think of other ways to do the problems. if you haven't done series solutions, it would be a lot to learn just to do these. so you probably aren't expected to do them that way
I solved the second one. In case someone has a similar problem, I've found this page Pauls Online Notes : Differential Equations - Euler Equations to be useful.
Euler type equations have polynomial type solutions.
Just substitute and get a parabola of , solve it and get two roots and (this is for second-order equations, for higher-order equations you get polynomial of degree ).
Then, the solution is
where and are arbitrary constants.
Note. Note that will decrease the power of by while the factor will increase it by , hence you will get for any .
This is why we search solutions of polynomial type.