Answer to Question #277369 in Complex Analysis for Aryaman

Question

The PDE 2∂

2

z

∂x

2

+2∂

2

z

∂y

2

+4∂

2

z

∂x∂y

=2x

2∂2z∂x2+2∂2z∂y2+4∂2z∂x∂y=2x,

is 

Answer 1

The PDE 

$2\frac{∂^2z}{∂x^2}+2\frac{∂^2z}{∂y^2}+4\frac{∂^2z}{∂x∂y}=2x^2\frac{∂^2z}{∂x^2}+2\frac{∂^2z}{∂y^2}+4\frac{∂^2z}{∂x∂y}=2x$

second order pde can be written in the form:

$az_{xx}+2bz_{xy}+cz_{yy}+dz_x+ez_y+fz=g(x,y)$

for $2\frac{∂^2z}{∂x^2}+2\frac{∂^2z}{∂y^2}+4\frac{∂^2z}{∂x∂y}=2x$ :

$b^2-ac=4-4=0$

this is parabolic equation

for $2x^2\frac{∂^2z}{∂x^2}+2\frac{∂^2z}{∂y^2}+4\frac{∂^2z}{∂x∂y}=2x$ :

$b^2-ac=4-4x^2$

then:

for $x=0$ :

this is parabolic equation

for $x<-2$ and $x>2$ :

$b^2-ac<0$

this is elliptic equation

for $-2<x<2$ :

$b^2-ac>0$

this is hyperbolic equation