Assume the following first order linear partial differential equation (PDE):
$$\sum_{i=1}^na_i(x_1,\ldots,x_n)\frac{\partial u}{\partial x_i}=1,$$
where the functions $a_i$ are analytic in $x_i$.
Question 1: Is it true that the corresponding homogeneous PDE
$$\sum_{i=1}^na_i(x_1,\ldots,x_n)\frac{\partial u}{\partial x_i}=0$$
always has $n-1$ functionally independent solutions $u_k$, $k=1,\ldots,n-1$?
Question 2: Is it true that the original PDE always has a particular solution?
Remarks:
- there are no boundary/initial conditions. I am interested in the homogeneous/particular solutions as such.
- I am aware that the solutions, if they exist, are not unique (since they can be combined to new solutions).
- I searched the literature, but I mainly found sources from physics where (i) existence was implicitly assumed, (ii) only specific PDEs where discussed (typically with $n=2$), or (iii) where the discussion focussed on the case where boundary/initial conditions were defined.
- The "best" I could find was the Cauchy–Kowalevski theorem, which however seems too general and which also assumes initial conditions. Also, it only guarantees local existence, and doesn't talk about homogeneous solutions (which makes sense in the nonlinear case).
- If existence cannot be guaranteed in general, I am interested in conditions where this is possible.