Consider a linear first order PDE $f(x)u_x + g(y)u_y = F(u), u = u(x,y)$, where $f(x)$, $g(y)$, $F(u)$ are analytic functions, is it safe to assume that all solutions to this PDE are analytic as well?
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Since $F(u)$ is in general non-linear, this equation is non-linear. I edited your title accordingly. Cheers! – Robert Lewis Nov 18 '17 at 18:46