$\displaystyle u_{xx}+u_{yy}+3u_x-4u_y+25u=0$

$\displaystyle u_{xx}=\omega_{\xi\xi}\cos^{2}{\alpha}-\omega_{\xi\eta}2\cos{\alpha}\sin{\alpha}+\omega_{ \eta\eta}\sin^{2}{\alpha}$

$\displaystyle u_{yy}=\omega_{\xi\xi}\sin^{2}{\alpha}+\omega_{\xi \eta}2\cos{\alpha}\sin{\alpha}+\omega_{\eta\eta}\c os^{2}{\alpha}$

$\displaystyle u_x=\omega_{\xi}\cos{\alpha}-\omega{\eta}\sin{\alpha}$

$\displaystyle u_y=\omega_{\xi}\sin{\alpha}+\omega{\eta}\cos{\alp ha}$

$\displaystyle \displaystyle\omega_{\xi\xi}(\cos^2{\alpha}+\sin^2 {\alpha})+\omega_{\eta\eta}(\cos^2{\alpha}+\sin^2{ \alpha})+\omega_{\eta\xi}(2\sin{\alpha}\cos{\alpha }-2\sin{\alpha}\cos{\alpha})+\omega_{\xi}(3\cos{\alp ha}-4\sin{\alpha})+\omega_{\eta}(-4\cos{\alpha}-3\sin{\alpha})+25\omega=0$

$\displaystyle \omega_{\xi\xi}+\omega_{\eta\eta}+\omega_{\xi}(3\c os{\alpha}-4\sin{\alpha})+\omega_{\eta}(-4\cos{\alpha}-3\sin{\alpha})+25\omega=0$

Since $\displaystyle \omega_{\xi\eta}$ is already gone, should I be working to eliminate $\displaystyle \omega_{\xi} \ \mbox{and} \ \omega_{\eta}\mbox{?}$