Solve the following differential equation: `(1+x^2)(dy)/(dx)-x=2tan^(-1)x`

Explanation

`(1+x^{2}) frac{d y}{d x}-x=2 tan ^{-1} x`

`
Rightarrow(1+x^{2}) frac{d y}{d x}=x+2 tan ^{-1} x`

`
Rightarrow d y={frac{x}{1+x^{2}}+(frac{2}{1+x^{2}}) tan ^{-1} x} d x`

`
Rightarrow d y={frac{1}{2} times frac{2 x}{1+x^{2}}+(frac{2}{1+x^{2}}) tan ^{-1} x} d x`


By Integrating, we obtain

`
int d y=int{frac{1}{2} times frac{2 x}{1+x^{2}}+(frac{2}{1+x^{2}}) tan ^{-1} x} d x`

`
Rightarrow y=frac{1}{2} int frac{2 x}{1+x^{2}} d x+2 int[frac{1}{1+x^{2}} tan ^{-1} x] d x`

`
Rightarrow y=frac{1}{2} log |1+x^{2}|+2 int[frac{1}{1+x^{2}} tan ^{-1} x] d x`


...