Question

if y = (tan-1 x)^2, prove that (1+x^2)^2 d2y/dx2+ 2x(1+x^2) dy/dx = 2

Answer 1

y = (tan⁻¹ x)² 
dy/dx  = 2 (Tan⁻¹ x) / (1+x²)
d²y/dx² = 2 [ (1+x²) * 1/(1+x²)  -  Tan⁻¹ x * (2x) ] / (1+x²)²
            = 2 [ 1 - 2 x Tan⁻¹ x ] /(1+x²)²

LHS =  (1+x²)² y'' + 2x (1+x²) y' 
        =  2 (1 - 2 x  Tan⁻¹ x)  + 2x (1+x²) * 2 Tan⁻¹ x * 1/(1+x²)
        = 2   
proved.

Answer 2

y =(tan^-1x)²
differentiate wrt x

dy/dx =2tan^-1x /(1+ x²)

(1 + x²)dy/dx = 2tan^-1x

again differentiate wrt x

( 1 + x²)d²y/dx² + ( 0 + 2x)dy/dx = 2/( 1+x²)

( 1 + x²)² d²y/dx² + 2x( 1 + x²)dy/dx = 2
hence proved