Question

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

Answer 1

$xlogx\frac{dy}{dx}+y=2logx$

divide throughout with $xlogx$

$\frac{dy}{dx}+\frac{y}{xlogx}=\frac{2}{x}$

$\frac{dy}{dx}+y\left(f\right(x\left)\right)=g\left(x\right)$

Linear differential

Integrating factor $e^{\int f\left(x\right)dx}$

$=e^{\int \frac{1}{xlogx}dx}$

$\left[\begin{array}{c}logx=t\\ \frac{1}{x}dx=dt\end{array}\right]$

$=e^{\int \frac{dt}{t}}=e^{lnt}=t$

$=logx$

$\therefore IF=logx$

$y\times \left(IF\right)=\int \left(IF\right)g\left(x\right)dx+c$

$ylogx=\int \frac{logx\left(2\right)}{x}dx+c$ $\left[\begin{array}{c}logx=p\\ \frac{1}{dx}dx=dp\end{array}\right]$

$ylogx=2\int pdp+c$

$ylogx=p^2+c$

$ylogx=(log{)}^2+c$

$y=logx+\frac{c}{logx}$.