# Canonical form of PDE

1. May 21, 2007

### 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?

2. Jun 2, 2007

### HallsofIvy

Staff Emeritus
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.