Question

Given that y=sin x ln (x+1),prove that

(d^2 y)/(dx^2 )+y=2cosx/(x+1)-sinx/(x+1)^2

Answer 1

Use the product rule:

if p=sin(x), q=ln(x+1)
y=p*q
dy/dx = (dp/dx)q + p(dq/dx)
=cos(x)ln(x+1)+sin(x)/(x+1)
and afterwards use the quotient rule for the second term:
d(u/v)/dx = [ v(du/dx)-u(dv/dx) ] / v²