I want to understand how (and if) I can use Sturm-Liouville machinery on the following simple example

$$ x^2 y''(x) + 2x y'(x) - y(x) = f(x) ~,$$

where $f(x)$ is some given function, and I am interested in the domain $x \in [0,1]$ with initial values given as $y(0)=0$ and $y'(0)=1$.

I understand how to get the solution by directly doing the Greens function method, but I would like to understand this example in the Sturm-Liouville approach.

First of all, given that I have an initial condition problem where $y(0)$ and $y'(0)$ are given, and not a boundary value problem, as is usually used for Sturm-Liouville, can I proceed? Is the formalism still applicable? Or is it really crucial to have conditions in the form

$$ a_1 y(0) + a_2 y'(0)=0, ~~\& ~~ b_1 y(1) + b_2 y'(1)=0 ~,$$

instead of the initial conditions form? I guess this form guarantees that the differential operator is Hermitian etc., but does the other case above necessarily exclude it?

If I am not able to use the Sturm-Liouville machinery above, is there an optimal way to pick a complete basis $e_n(x)$ on this $L^2(0,1)$ space, so that I can decompose my Greens function $G(x,x')$, which would be a solution of the equation

$$ x^2 G''(x',x) + 2x G'(x',x) - G(x',x) = \delta(x'-x) ~, $$

(with corresponding i.c.), in this $e_n(x)$ basis as

$$ G''(x',x) = \sum_{n} a_n(x')e_n(x) ~?$$

My goal is to be able to write the solutions in the form

$$y(x) = \int_0^1 G(x,x') f(x')dx' ,$$

for which, if I also decompose $f(x)$ in the $e_n(x)$ basis as $f(x) = \sum_m c_m e_m(x)$, it would simply give me

$$y(x) = \sum_n c_n a_n(x) ~ .$$

Is this feasible or am I missing something here?


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