For the differential equation in given question find the general solution.

$\frac{d y}{d x} = (1 + x^{2}) (1 + y^{2})$

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Solution

Given, $\frac{d y}{d x} = (1 + x^{2}) (1 + y^{2})$
On separating the variables, we get
$\frac{d y}{1 + y^{2}} = d x (1 + x^{2}) \Rightarrow \frac{d y}{1 + y^{2}} = 1 d x + x^{2} d x$
Integrating, we get $\int \frac{d y}{1 + y^{2}} = \int d x + \int x^{2} d x \Rightarrow t a n^{- 1} y = x + \frac{x^{3}}{3} + C$
which is required general solution.

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