Question

Solve:
$x\frac{dy}{dx}+2y=x^2\log x$

Answer 1

$x.\frac{dy}{dx}+2y=x^{2}logx$

$\Rightarrow \frac{dy}{dx}+\frac{2}{x}.y=x\log x$ -------(1)

$\because$ equation (1) is linear equation so,

$I.F=\underset{e}{\int }\frac{2}{x}.dx=e^{2logx}=x^2.$

solution (1)

$y.x^2=\int \underset{II}{x^3}.\underset{I}{logx}.dx$

$=logx.\int x^3.dx-$

$=logx.\int x^3.dx-\int [\frac{d}{dx}.logx\int x^3.dx].dx$

$logx.\frac{x^4}{4}-\int \frac{1}{x}.\frac{x^4}{4}.dx$

$y.x^2=\frac{x^4}{4}.logx-\frac{1}{4}.\int x^3.dx$

$y.x^2=\frac{x^4}{4}.logx-\frac{1}{4}.\frac{1}{4}.x^4+C$

$y=\frac{x^2}{4}logx-\frac{1}{16}.x^2+C.x^{-2}$