TITLE: Linear Second order ODE with oscillating solutions
QUESTION [0 upvotes]: I encountered the following second-order ODE while tutoring recently, and struggled with the proper approach:
$x^2y''+2xy'+\alpha y = 0$
The problem is: for which values of $\alpha$ do solutions oscillate as $x\to \infty$?
Eventually, we gave up and asked Wolfram Alpha for a solution. Based on what we saw there, we were able to reverse engineer a method for solving it: assume that $y(x)=x^r$, plug this assumption into the original equation, and solve for $r$. It turns out there is a $\sqrt{1-4\alpha}$ term in the resulting solution ($r=\frac12(-1\pm\sqrt{1-4\alpha})$), so you get osciallating solutions if $\alpha>\frac14$. Super.
My question is this: what kind of intuition should guide one's approach to this problem? How might I have known which form of solution to look for? What is the usual approach for a problem like this? There's kind of a nice symmetry to the original equation - how can I use that to guide me towards a solution?
Thanks in advance for any help.

REPLY [1 votes]: The thing here is that with $y=x^r$, when apply the derivative operator $k$ times and then multiply by $x^k$, you keep the same power. So there are chances to find a solution this way.
Alternatively, you can try with the change of variables $t = \log x$ to get back to constant coefficients.
{"set_name": "stack_exchange", "score": 0, "question_id": 987187}