If y = sin(m cos^-1 x), then show that (1 - x^2) d^2 y/dx^2 - x dy/dx + m^2 y = 0

Question

If y = sin(m cos^-1 x), then show that \((1 - x^2) \frac{d^2 y}{dx^2} - x\frac{dy}{dx}\) \(+\, m^2 y = 0\)

(1 - x²) d²y/dx² - x dy/dx + m²y = 0

Answer 1

y = sin(m cos^-1 x)

sin^-1y = m cos^-1 x

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

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