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​ ​ 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.​

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  • $\begingroup$ Please try to answer my question. $\endgroup$
    – Adm
    Feb 11, 2016 at 5:50

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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 $v = 0$, $v = y$.

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  • $\begingroup$ Hi and welcome to MSE. Please use mathjax to format your answer. $\endgroup$
    – Surb
    Jan 16, 2017 at 11:03

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