(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I must solve [itex]x(1-x)y''+4y'+2y=0[/itex].

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 "[itex]f(tx,ty)=t^nf(x,y)[/itex]", 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!

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# Homework Help: 2nd order DE+ question on theory

Can you offer guidance or do you also need help?

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