# 2nd order DE+ question on theory

1. Oct 9, 2011

### fluidistic

1. The problem statement, all variables and given/known data
I must solve $x(1-x)y''+4y'+2y=0$.
According to Boyce and Di Prima's book, Paul's note on DE on the internet and the notes of my professor, such a DE is homogeneous. However if I use the definition of homogeneous "$f(tx,ty)=t^nf(x,y)$", I don't find that's it's homogeneous of any order n. Why is this? I had to assume that if y transforms into ty, then y' transforms into t^2y' and y'' transforms into t^3y''. Seems like it's wrong but I don't understand why.
Back to the problem, I notice that x(1-x), the term in front of y'' isn't non zero for x=0 and x=1, thus I can have troubles if I divide the whole equation by x(1-x) (if I'm right, I can't assume there will be only 1 solution to the DE, but there might be an infinity even given initial conditions).

2. Relevant equations
I tried to figure out in books/Internet but I don't understand well if they give a general solution/method to find the general solution to the DE...
For instance, in Paul's notes, I think that the page would be http://tutorial.math.lamar.edu/Classes/DE/FundamentalSetsofSolutions.aspx but I don't know how to proceed in my example. It seems like all the guys who write about DE assume that I already know solutions to the DE but here I don't know any solution so I don't know how to start.

3. The attempt at a solution
Stuck on starting. I would like a reference to a book or better, a page on the internet. (I'm not asking a solution here :) ).
Thanks for any help!