More powerful simplifications are possible in 2D.

In the initial independent coordinates x,y:

a u_xx + 2 b u_xy + c u_yy + d = 0

In the new independent coordinates

The new coefficients may be found by writing out the transformation formulae from the introduction for the two-dimensional case, and are:

The trick now is to demand that a', b' and c' are as simple as possible and from that compute what the required and are.