Answer to Question #140577 in Differential Equations for Nikhil

Question

The differential equation
(1-x^2)(∂^2z/∂x^2 -2xy[∂^2z/(∂x.∂y)] + (1-y^2)(∂^2z/∂y^2) + 2(∂z/∂x) +3y(∂z/∂y)=0, is elliptic for (x,y) outside the circle x^2 + y^2=1
True or false with correct explanation

Answer 1

Given differential equation is

$(1-x^2)(∂^2z/∂x^2) -2xy[∂^2z/(∂x.∂y)] + (1-y^2)(∂^2z/∂y^2) + 2(∂z/∂x) +3y(∂z/∂y)=0$ .

Compare it with

$A(∂^2z/∂x^2)+B[∂^2z/(∂x.∂y)] + C(∂^2z/∂y^2) + D(∂z/∂x) +E(∂z/∂y)=0$ , we have

$A= 1-x^2, B = -2xy, C =1-y^2$ .

Now, $B^2-4AC = 4x^2 y^2 - 4(1-x^2)(1-y^2) = 4(x^2+y^2-1)$

For (x,y) outside the circle $x^2 + y^2=1$ , we have $x^2 + y^2 > 1$ .

$\implies B^2-4AC > 0$ .

Hence, the given differential equation is Hyperbolic equation.

Thus, given statement is false.