Question

If y = (tan^-1 x)², show that

`(1+x^2)^2(d^2y)/dx^2+2x(1+x^2)dy/dx-2=0` 

Answer 1

y = tan ^-1x 

y₁ = `1/(1+"x"^2)`

y₂ = `(d(1+x^2)^(-1))/"dx" = -1/(1+x^2)^2 xx ("2x")`

y₂ = `(-2x)/(1+x^2)^2`

y₂ (1+x²) = -2x `(1/(1+x^2))`

y₂ (1+x²) = -2 xy₁ 

y₂ (1+x²) + 2xy₁ = 0