Question

If \(y^{\frac{1}{m}} + x^{\frac{1}{m}} = 2x\) then

• A. \((x^2 + 1) \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - m^2 y = 0\)
• B. \((x^2 - 1) \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - m^2 y = 0\)
• C. \((x^2 + 1) \frac{d^2 y}{dx^2} - x \frac{dy}{dx} + m^2 y = 0\)
• D. \((x^2 + 1) \frac{d^2 y}{dx^2} - x \frac{dy}{dx} - m^2 y = 0\)

Hint:

The Lande \( g \)-factor determines the energy shift in a magnetic field for fine-structure states.

Answer 1

The Correct Option is C

Let \(z = y^{1/m} \Rightarrow z + x^{1/m} = 2x\)
Differentiate twice and rearrange using chain rule. Eventually, you’ll reach option (c) as satisfying the condition.