Formula: – Given: – y = x n (acos(logx) + bsin(logx)) y = ax n cos(logx) + bx n sin(logx) = x n (na + b)[(n – 1) cos(logx) – sin (logx) ] + (bn – a) x n [(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 + n 2 )x n cos(logx) + bx n (1 + n 2 )sin(logx)

If , prove that

Formula: –

Given: –

y = xⁿ(acos(logx) + bsin(logx))

y = axⁿcos(logx) + bxⁿsin(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)