$y^2r-x^2t=0$
where $r=d^2z/dx^2$
$t=d^2z/dy^2$
I know a way to find characteristics in single order equation. But not in second order. How do we proceed here ?
2026-03-27 07:48:11.1774597691
Finding Characteristics of second order Partial differential Equation
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Can we find $a,b$ such that there exist solutions $z=Z(ax+by)$? If so, then substitute into pde, $(y^2a^2-x^2b^2)Z"=0$. and $a/b=\pm x/y$. These give characteristic lines along which $a\partial_xz+b\partial_yz=0$. There are more sophisticated ways to do it.
P.S.(whisper) $r$ and $t$ are partials.