Originally Posted by

**dwsmith** Show that the only second-order linear homogeneous equation with constant coefficients in x and y whose form is unchanged by all rotations of axes is $\displaystyle u_{xx}+u_{yy}+ku=0$.

$\displaystyle \omega_{\xi\xi}(\cos^2+\sin^2)+\omega_{\eta\eta}(\ cos^2+\sin^2)+k\omega=0\Rightarrow\omega_{\xi\xi}+ \omega_{\eta\eta}+k\omega=0\Rightarrow u_{xx}+u_{yy}+ku=0$

Showing this doesn't change is the easy part (see above), but how do I show it is the only second-order linear?

Thanks.