Solve the following differential equation: `(1+y^2)tan^(-1)dx+2y(1+x^2)dy=0`

Explanation

We have,

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`(1+y^{2}) tan ^{-1} x d x+2 y(1+x^{2}) d y=0 `

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

`rightarrow quad frac{tan ^{-1} x d x}{1+x^{2}}=-frac{2 y}{1+y^{2}} d y `

`rightarrow quad quad int frac{tan ^{-1} x}{1+x^{2}} d x=-int frac{2 y}{1+y^{2}} d y `

`rightarrow quad quad frac{1}{2}(tan ^{-1} x(tan ^{-1} x)^{prime} d x=-int frac{(1+y^{2})^{prime}}{1+y^{2}} d y. `

`rightarrow quad frac{1}{2}(tan ^{-1} x)^{2}+log (1+y^{2})+C `

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