Transforming a Second-Order PDE into Canonical Form: Tips and Techniques

[r4nd0m]

How do I transform a second-order PDE with constant coefficients into the canonical form?

I tried to solve this problem:
u_xx + 13u_yy + 14u_zz - 6u_xy + 6u_yz + 2u_xz -u_x +2u_y = 0

I wrote the bilinear form of the second order derivatives and diagonalized it. I found out that it is a hyperbolic equation. Now the problem is how to write it into the canonical form.

What I tried is I wrote it as:
u_aa + u_bb + u_cc + ...(first order derivatives) = 0
where a,b,c are the new variables (in which the matrix is diagonal) and computed the first order derivatives.
Is this a good approach or something else should be done?

 

[HallsofIvy]

Do you know how to convert a general conic section equation to its "cononical form"? It's really the same method. Replace the partial derivatives with x, x², y, y², etc. and convert that equation. Then change back to the partial derivatives.