How to find canonical form of $u_{xx}+36u_{xy}+243u_{yy}=0$

Question

I need to find the canonical form of $$u_{xx}+36u_{xy}+243u_{yy}=0$$ and find the general solution but I have no idea how to find the canonical form, I am new to the subject in general, so any help would be appreciated.

Answer 1

Hint.

$x^2+36x y +243 y^2 = (x+9y)(x+27 y)$

then

$$ (\partial_x+9\partial_y)(\partial_x+27\partial_y)u = 0 $$

so we can proceed as

$$ (\partial_x+9\partial_y)v=0\\ (\partial_x+27\partial_y)u = v $$

Also can be easily solved after a coordinates change. Making the change of coordinates $$ \xi = a_1 x + b_1 y\\ \eta = a_2 x+ b_2 y $$

we get at

$$ (a_2^2+36a_2b_2+243 b_2^2)u_{\eta\eta}+2(a_1a_2+18a_2b_1+243b_1b_2)u_{\xi\eta}+(a_1^2+36a_1b_1+243 b_1^2)u_{\xi\xi}=0 $$

and now choosing

$$ a_1 = -9b_1\\ a_2 = -27b_2 $$

we finally arrive at the simpler version

$$ u_{\xi\eta}(\xi,\eta)=0 $$

and thus

$$ u(\xi,\eta) = \phi(\xi)+\theta(\eta) $$