Question

The differential equation \[ 2\frac{\partial^2 z}{\partial x^2} + 5\frac{\partial^2 z}{\partial x\partial y} + 2\frac{\partial^2 z}{\partial y^2} + 7\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} =0 \] is classified as

• A. Parabolic
• B. Elliptic
• C. Hyperbolic
• D. Clairaut's equation

Hint:

For PDE classification, remember only one formula: \[ D=B^2-4AC. \] Positive discriminant gives Hyperbolic, zero gives Parabolic, and negative gives Elliptic.

Answer 1

The Correct Option is C

Concept: A second-order partial differential equation \[ A z_{xx}+B z_{xy}+C z_{yy}+\cdots=0 \] is classified using the discriminant \[ D=B^2-4AC. \] Classification: \[ D>0 \Rightarrow \text{Hyperbolic} \] \[ D=0 \Rightarrow \text{Parabolic} \] \[ D<0 \Rightarrow \text{Elliptic} \]

Step 1: Identify the coefficients.
Comparing with \[ Az_{xx}+Bz_{xy}+Cz_{yy}=0, \] we get \[ A=2, \qquad B=5, \qquad C=2. \]

Step 2: Compute the discriminant.
\[ D=B^2-4AC. \] \[ D=5^2-4(2)(2). \] \[ D=25-16. \] \[ D=9. \]

Step 3: Classify the PDE.
Since \[ D=9>0, \] the equation belongs to the hyperbolic category. \[ \boxed{\text{Hyperbolic}} \]