If y = (tan⁻¹x)² show that (x²+1)²(d²y/dx²) + 2x(x²+1)(dy/dx) = 2

Given y = (tan⁻¹x)² —(1)
Differentiating w.r.t to x, we get
dy/dx = 2tan⁻¹x * 1/(1+x²) —(2)
or (1+x²)y′ = 2tan⁻¹x
Again differentiating with w.r.t to x, we get
(1+x²)(dy′/dx) + y′d(1+x²)/dx = 2 * 1/(1+x²)
⇒(1+x²).y″ + y′.2x = 2/(1+x²)
⇒(1+x²)².y″ + y′.2x(1+x²) = 2
Therefore,
⇒(1+x²)²(d²y/dx²) + 2x(1+x²)(dy/dx) = 2