Partial Differential Equation

Calculus Level pending

x ∂ 2 u ∂ x 2 + y ∂ 2 u ∂ y 2 = 0 x\frac{\partial^2 u}{\partial x^2} + y\frac{\partial^2 u}{\partial y^2} = 0

Above partial differential equation is

Hyperbolic for x > 0 x>0 , y > 0 y>0 Elliptiic for x < 0 x<0 , y > 0 y>0 Elliptiic for x > 0 x>0 , y < 0 y<0 Hyperbolic for x > 0 x>0 , y < 0 y<0

Ravneet Singh by Ravneet Singh , 30, India

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1 solution

Tom Engelsman
May 4, 2017

Taking a second-order partial differential equation of the standard form:

A u x x + 2 B u x y + C u y y + D u x + E u y + F = 0 Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0

we have the following criteria for classification of this PDE:

B 2 − A C < 0 ⇒ e l l i p t i c a l B^2 - AC < 0 \Rightarrow elliptical ; B 2 − A C = 0 ⇒ p a r a b o l i c B^2 - AC = 0 \Rightarrow parabolic ; B 2 − A C > 0 ⇒ h y p e r b o l i c B^2 - AC > 0 \Rightarrow hyperbolic .

The original PDE has the parameters: A = x A = x , B = D = E = F = 0 B = D = E = F = 0 ; C = y C = y . It will be elliptical iff x , y x,y are nonzero & identical in sign. Conversely, it will be hyperbolic iff they are non-zero & opposite in sign. Choice D is the only satisfactory answer.

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