In a set of practice problems I was asked this:
If $\gamma - \beta \geq -\frac{1}{4}$ then for any $A,B$ $\exists!$ extremal for
$\inf\{\int^b_a x^2y'^2+2\beta x yy' + \gamma x^2\}$
I am trying to solve this issue, so i took the first obvious step and computed the E-L equation getting:
$$2x^2y'' + 4xy' + 2(\beta - \gamma)y = 0$$
At this point I am already screwed because I don't know how to solve a second order ODE with non linear coefficients.
I decided to pretend the coefficients were constant and calculate the root of the associated polynomial.
This didn't give me something useful and I don't think it's the correct approach anyway.
How should I tackle this problem?