If log(x + √(x^2 + a^2))m, show that (x^2 + a^2)d^2 y/dx^2 + x dy/dx = 0

Question

If y = log \((x + \sqrt{x^2 + a^2})^m,\) show that \((x^2 + a^2) \frac{d^2 y}{dx^2} + x \frac{dy}{dx} = 0\)

y = log(x + √(x² + a²))^m, show that (x² + a²)d²y/dx² + x dy/dx = 0

Answer 1

y = log(x + √(x² + a²))^m 

 y = log \((x + \sqrt{x^2 + a^2})^m\) = m log \((x + \sqrt{x^2 + a^2})\)

Differentiating both sides w.r.t. x, we get

Cancelling 2 dy/dx throughout, we get