My subject about the canonical form of PDE. I had many exercises to do and they were fine, but I'm stuck with this one: $$U_{xx}-yU_{xy}+xU_x+yU_y+u=0$$ So first we have to calculate $B-4AC=y^2-4$. I couldn't determine what ether hyperbolic or elliptic or parabolic?
Could you help me to get the canonical form? Thanks in advance.
This equation is hyperbolic.
First you look at only the second order stuff to get $u_{xx} - y u_{xy} = 0$.
This gives the auxiliary equation $v^2 - yv = 0$, which is $v(v-y) = 0$, giving two real distinct roots of $v = 0$, $v = y$.