Question:
$\frac{d y}{d x}=\left(1+x^{2}\right)\left(1+y^{2}\right)$
Solution:
The given differential equation is:
$\frac{d y}{d x}=\left(1+x^{2}\right)\left(1+y^{2}\right)$
$\Rightarrow \frac{d y}{1+y^{2}}=\left(1+x^{2}\right) d x$
Integrating both sides of this equation, we get:
$\int \frac{d y}{1+y^{2}}=\int\left(1+x^{2}\right) d x$
$\Rightarrow \tan ^{-1} y=\int d x+\int x^{2} d x$
$\Rightarrow \tan ^{-1} y=x+\frac{x^{3}}{3}+\mathrm{C}$
This is the required general solution of the given differential equation.
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