Solve: d y d x +2 x y= x ^2 - Vidyadip

Question

Solve: $\frac{ d y}{ d x}+\frac{2}{x} y= x ^2$

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Answer

$\frac{ d y}{ d x}+\frac{2}{x} y= x ^2$
The given equation is of the form
$ \frac{ d y}{ d x}+ P y= Q .$
$\text { where } P =\frac{2}{x} \text { and } Q = x ^2$
$\therefore \text { I.F. }= e ^{\int^{ P d x}}$
$= e ^{\int \frac{2}{x}} d x$
$= e ^{2 \log x}$
$= e ^{\log x^2}$
$= x ^2 $
$\therefore$ Solution of the given equation is
$ y(\text { I. F. })=\int Q (\text { I.F. }) d x+ c _1$
$\therefore y \cdot x^2=\int x^2 \cdot x^2 d x+ c _1$
$\therefore y x^2=\int x^4 d x+ c _1$
$\therefore yx ^2=\frac{x^5}{5}+ c _1$
$\therefore 5 x ^2 y = x ^5+ c _1 \text { where } c =5 c _1 $

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