Let 
$$
(\star) \;\;\;\;\;\;\;\;\;\;\;\; y''+p(x)y'+q(x)y=0,
$$ be a homogeneous linear ODE of order $2$ with $p(x)$ and $q(x)$ complex valued analytic functions in a small neighbourhood of $0$.


**Q:** Is there a "closed formula" involving only


(1) elementary functions


(2) elementary arithmetic operations


(3) Integrals (line integrals)


(4) the functions $p(x)$ and $q(x)$


which gives always a non-zero solution to the ODE $(\star)$ in a neighborhood of $0$?


P.S. Of course if $(\star)$ has (at most) a regular singularity at $0$, then using power series (which is not part of the toolkit allowed in my question)
it is possible (in principle) to write a non-zero solution.


**added:** For example, if $y\_1$ is a non-zero solution to $\star$, one may obtain another solution $y\_2$ of $(\star)$ (linearly independent of $y\_1$) using Lagrange's method (the "variation" of the constant) which gives using indefinite integrals
$$
y\_2(x)=y\_1(x)\int (y\_1(x))^{-2}e^{-\int p(x)dx} dx
$$


Note that $q(x)$ does not appear explicitly in the formula but it is hidden in $y\_1(x)$.
So one may find a second solution if one has an "input" solution $y\_1$ 



closed integral formula for a non-zero solution of a homogeneous linear ODE of order 2
The answer is basically 'no', there is no 'elementary method' involving elementary operations and quadrature (i.e., finding antiderivatives of known holomorphic functions) that will give you a solution to the general second order linear equation with variable coefficients. 


For a glimpse at why (an explanation is too complicated to summarize here), you might consult the delightful book by Michio Kuga, *Galois' Dream: Group Theory and Differential Equations*.


