Question
The general solution of the differential equation $x d y-\left(1+x^2\right) d x=d x$ is :

Answer

Given differential equation is
$x d y-\left(1+x^2\right) d x =d x$
$\Rightarrow x d y =d x+\left(1+x^2\right) d x$
$ =\left(2+x^2\right) d x$
$\Rightarrow d y =\left(\frac{2}{x}+x\right) d x$
Integrating both sides, we get
$\int d y=\int\left(\frac{2}{x}+x\right) d x$
$\Rightarrow y=2 \log x+\frac{x^2}{2}+C$

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