If y = x^n {a cos(log x) + b sin (log x)}, prove that x^2 d^2y/dx^2 +(1 - 2n) dy/dx + (1+n^2)y=0

Question

If y = xⁿ {a cos(log x) + b sin (log x)}, 

prove that x² \(\frac{d^2y}{dx^2}+(1-2n)\frac{dy}{dx}+(1+n^2)y=0\)

Answer 1

Formula: –

Given: –

y = xⁿ (acos(logx) + bsin(logx))

⇒ y = axⁿ cos(logx) + bxⁿ sin(logx)

= xⁿ (na + b)[(n – 1) cos(logx) – sin (logx) ] + (bn – a) xⁿ [(n – 1) sin(logx) + cos(logx)] + (1 – 2n)x^{n – 1} cos(logx)(na + b) + (1 – 2n)x^{n – 1}sin(logx)(bn – a) + a(1 + n²)xⁿcos(logx) + bxⁿ(1 + n²)sin(logx)