If y = sin (log x), then prove that x^2(d^2y)/dx^2+dy/dx+y = 0. ​

Question

If y = sin (log x), then prove that \(x^2\frac{d^2y}{dx^2}+\)\(\frac{dy}{dx}+y=0\).

Answer 1

Given,

y = sin (log x)

= RHS

Answer 2

y = sin(logx)

dy/dx = cos(logx) / x

d²y/dx² = [-sin(logx) - cos(logx) ] / x²

X² . d²y/ dx² = -y -cos(logx)

X² . d²y/ dx² = -y - x dy/dx

X² d²y/dx² + y + x dy/dx = 0

Hence proved.