I'm stuck with these three ODEs:
Nothing I tried worked, could you please help me? A hint on which method to try would be really great.
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.
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.