Question

If \[ \theta = r e^{-x^2}, \] then for what value of \( n \), the following holds: \[ \frac{1}{2} \left( \frac{\partial^2 \theta}{\partial x^2} - \frac{\partial^2 \theta}{\partial r^2} \right) = \frac{\partial \theta}{\partial x}. \]

• A. \( \frac{1}{2} \)
• B. 0
• C. 1
• D. \( \frac{-3}{2} \)

Hint:

Solve partial differential equations by computing the necessary derivatives and comparing both sides of the equation.

Answer 1

The Correct Option is D

The value of n is determined by applying the given partial derivatives. After performing the calculations, we find that n = \( \frac{-3}{2} \)satisfies the given equation.