1.5.3.1 Solution 2dcanel-a

Question:

Convert the equation

$$10 u_{xx} + 6 u_{xy} + 2 u_{yy} = u_x + x + 1$$

to two-dimensional canonical form.

Using rotation and stretching of the coordinates you would get

$$u_{\xi\xi} + u_{\eta\eta} = \frac{3}{\sqrt{110}} u_{\xi} - \... ...frac{3\sqrt{11}}{\sqrt{10}} \xi - \frac{1}{\sqrt{10}} \eta + 1$$

Do you get the same equation? Should you? Comment.

Answer:

You get

$$u_{\xi\xi} + u_{\eta\eta} = \frac{3}{11} u_{\xi} + \frac{1}{\sqrt{11}} u_{\eta} + \frac{100}{11\sqrt{11}} \eta + 1$$

You do not necessarily get the same result. If you rotate the coordinate system, the Laplacian stays the same, but the right hand side changes.