[SOLVED] Linear second order, homogeneous, constant coefficients, unchanged by all rotations

Quote:

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 $u_{xx}+u_{yy}+ku=0$ .

$\omega_{\xi\xi}(\cos^2+\sin^2)+\omega_{\eta\eta}(\ cos^2+\sin^2)+k\omega=0\Rightarrow\omega_{\xi\xi}+ \omega_{\eta\eta}+k\omega=0\Rightarrow u_{xx}+u_{yy}+ku=0$
Showing this doesn't change is the easy part (see above), but how do I show it is the only second-order linear?

Thanks.

Start with

$a u_{xx} + b u_{xy} + c u_{yy} + du_x + e u_y + f u = 0$ .

Introduce your change of variables and require that

$a \omega_{\xi \xi} + b \omega_{\xi \eta} + c \omega_{\eta \eta} + d \omega_{\xi} + e \omega_{\eta} + f u = 0$ .