Question

Particular integral of the partial differential equation \( \frac{\partial^2 z}{\partial x^2} - 2 \frac{\partial^2 z}{\partial x \partial y} + \frac{\partial^2 z}{\partial y^2} = 2x \cos y \) is

• A. \( x \cos y + 2 \sin y \)
• B. \( -2(x \cos y + 2 \sin y) \)
• C. \( 2 \cos y + x \sin y \)
• D. \( 2 \cos y + \sin y \)

Hint:

Use operator methods for PDEs with constant coefficients. Recognize forms like \( (D - D')^2 \) and apply inverse operators accordingly.

Answer 1

The Correct Option is B

We simplify the operator: \[ \frac{\partial^2 z}{\partial x^2} - 2 \frac{\partial^2 z}{\partial x \partial y} + \frac{\partial^2 z}{\partial y^2} = (D - D')^2 z = 2x \cos y \] Now solve using operator method: \[ (D - D')^2 z = 2x \cos y \] Let \( z = P.I. = -2(x \cos y + 2 \sin y) \), as derived from standard methods of solving linear PDEs with constant coefficients.