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 neighbourhood of $0$?

P.S. Of course if $(\star)$ has an regular singularity at $0$ the using power series 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 $y_1$

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 $y_1$

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 neighbourhood of $0$?

P.S. Of course if $(\star)$ has an regular singularity at $0$ the using power series it is possible (in principle) to write a non-zero solution.

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 neighbourhood of $0$?

P.S. Of course if $(\star)$ has an regular singularity at $0$ the using power series 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 $y_1$