Question

The partial differential equation \[ \frac{\partial^2 u}{\partial x^2} + 4 \frac{\partial^2 u}{\partial x \partial y} + 2 \frac{\partial^2 u}{\partial y^2} = 0 \] is \_\_\_\_\_\_.

• A. elliptic
• B. hyperbolic
• C. parabolic
• D. of mixed type

Hint:

To classify partial differential equations, calculate the discriminant \( D = B^2 - AC \). If \( D>0 \), the equation is hyperbolic.

Answer 1

The Correct Option is B

The given partial differential equation is of the form: \[ A \frac{\partial^2 u}{\partial x^2} + 2B \frac{\partial^2 u}{\partial x \partial y} + C \frac{\partial^2 u}{\partial y^2} = 0 \] For classification, we calculate the discriminant \( D \): \[ D = B^2 - AC \] In this case, \( A = 1 \), \( B = 2 \), and \( C = 2 \), so: \[ D = 2^2 - (1)(2) = 4 - 2 = 2 \] Since \( D>0 \), the equation is hyperbolic.