Question

If y=log(x+√x^2+1), then prove that (x^2+1)d^2y/dx^2+xdy/dx=0

Answer 1

I have applied substitution method to solve this problem.

Take

x=tanβ

x²+1= tan²β+1= sec²β

dx/dβ = sec²β

y=log(tanβ+√tan²β+1)

y=log(tanβ+secβ)

dy/dβ = (1/tanβ+secβ).(sec²β+secβ.tanβ)

dy/dβ = (1/tanβ+secβ). secβ(secβ+tanβ)

dy/dβ = secβ

Now,

dy/dx=(dy/dβ).(dβ/dx)

dy/dx=secβ.(1/ sec²β)

dy/dx=1/ secβ

dy/dx= cosβ

d²y/dx² = - sinβ. (dβ/dx)

d²y/dx² = - sinβ.(1/sec²β)

(x²+1)(d²y/dx²)+x(dy/dx)

= sec²β.( - sinβ.(1/sec²β))+tanβ.cosβ

= ( - sinβ)+(sinβ/cosβ).cosβ

= (- sinβ)+sinβ

=0