For an ODE like $$ f''(x) + p(x)f'(x) + q(x)f(x) = 0\ , $$ one expects two linear independent solutions.
Now consider a system of second order linear PDEs, like $$ \begin{pmatrix} \partial_x^2 f(x, y)\\ \partial_x\partial_y f(x, y)\\ \partial_y^2 f(x, y) \end{pmatrix} = P(x, y) \begin{pmatrix} f(x, y)\\ \partial_x f(x, y)\\ \partial_y f(x, y)\\ \end{pmatrix} \ , $$ where $P(x,y)$ is a matrix of rational functions of $x, y$, and suppose we are looking for solutions of $f(x,y)$ in $x, y$-series. Is there a way to see how many linear independent solutions there are?
Is it possible to generalize to $n$-variate function $f$ where all its second order derivatives $\partial_i \partial_j f$ are determined linearly (with rational function of $x_1, ..., x_n$ as coefficients) by $\partial_i f$ and $f$?