2nd order DE+ question on theory

[fluidistic]

Homework Statement

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).

Homework 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.

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!

 

[mighty2000]

Thank you for sharing your question with us. I understand your confusion about the homogeneity of the given differential equation. Let me try to explain why it is not homogeneous of any order.

First, let's review the definition of a homogeneous differential equation. A differential equation of the form y'' + p(x)y' + q(x)y = 0 is said to be homogeneous if both p(x) and q(x) are functions of x only (i.e. they do not depend on y). In your equation, x(1-x) is a function of x only, but the term y'' is not. This means that your equation is not homogeneous.

You are correct in noting that if y transforms into ty, then y' transforms into t^2y' and y'' transforms into t^3y''. This is because of the chain rule in differentiation. However, this does not make the equation homogeneous. In fact, if you plug in ty for y and t^3y'' for y'' into the original equation, you will see that it does not simplify to t^n times the original equation. This is another indication that the equation is not homogeneous.

Now, regarding your question about finding a general solution to the equation, I would recommend looking into the method of variation of parameters. This is a technique for solving non-homogeneous linear differential equations like the one you have. You can find more information and examples of this method in most differential equations textbooks, such as Boyce and Di Prima. Additionally, you can find online resources that explain the method, such as this page from the MIT OpenCourseWare website: https://ocw.mit.edu/courses/mathema...ns-spring-2010/readings/MIT18_03S10_notes.pdf

I hope this helps you in your understanding and further exploration of this problem. Best of luck with your studies.