Question #277369

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 


1
Expert's answer
2021-12-09T17:25:32-0500

The PDE 


22zx2+22zy2+42zxy=2x22zx2+22zy2+42zxy=2x2\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:

azxx+2bzxy+czyy+dzx+ezy+fz=g(x,y)az_{xx}+2bz_{xy}+cz_{yy}+dz_x+ez_y+fz=g(x,y)


for 22zx2+22zy2+42zxy=2x2\frac{∂^2z}{∂x^2}+2\frac{∂^2z}{∂y^2}+4\frac{∂^2z}{∂x∂y}=2x :


b2ac=44=0b^2-ac=4-4=0

this is parabolic equation



for 2x22zx2+22zy2+42zxy=2x2x^2\frac{∂^2z}{∂x^2}+2\frac{∂^2z}{∂y^2}+4\frac{∂^2z}{∂x∂y}=2x :


b2ac=44x2b^2-ac=4-4x^2


then:

for x=0x=0 :

this is parabolic equation


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

b2ac<0b^2-ac<0

this is elliptic equation


for 2<x<2-2<x<2 :

b2ac>0b^2-ac>0

this is hyperbolic equation


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