2.24, §2 Solution

3 u_xx - 2 u_xy + 2 u_yy - 2 u_yz + 3 u_zz + 12 u_y - 8 u_z = 0

Identify the matrix:

To find the new coordinates (transformation matrix), find the eigenvalues and eigenvectors of A:

Hence , , .

For ,

For ,

For ,

The new equation is:

However, that still contains the old coordinates in the first order terms. Use the transformation formulae and total differentials to convert the first order derivatives:

Hence in the rotated coordinate system, the PDE is:

This could be reduced further by stretching the coordinates. If

then

Note that tall that is left in the second order derivative terms is the sign of the eigenvalues. Now you know why we classify based on the sign of the eigenvalues!

You could further set and choose a, b, and c to get rid of the first order derivatives. You need:

Then