The general solution of the differential equation x d y- (1+x^2 )… - Vidyadip

MCQ

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

A
$y=2 x+\frac{x^3}{3}+C$
B
$y=2 \log x+\frac{x^3}{3}+C$
C
$y=\frac{x^2}{2}+C$
D
$y=2 \log x+\frac{x^2}{2}+C$

✓

Answer

Given differential equation is
$x d y-\left(1+x^2\right) d x=d x$
$\Rightarrow x d y=d x+\left(1+x^2\right) d x$
$=\left(2+x^2\right) d x$
$\Rightarrow \quad d y=\left(\frac{2}{x}+x\right) d x$
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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