The PDE $$ 1 + (\partial f/\partial y)^2 + (\partial f/\partial x)^2 - C \bigl(\partial^2 f/ \partial x^2 \bigr)^2 = 0 $$ for a function $f$ of two real variables $x$ and $y$ and a real parameter $C > 0$ is, for instance, solved by $$ f(x, y) = \cosh(x) $$ if $C= 1$. However, that solution does not depend on $y$. Is there an ansatz to find more/all solutions, in particular ones with non-vanishing derivative by $y$? It seems this PDE should be in a well-studied class.
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