Question

Find integrating factor for the differential equation $x\log \left(x\right)\frac{dy}{dx}+y=2\log \left(x\right)$

Answer 1

Lets take the given differential equation
$x\log \left(x\right)\frac{dy}{dx}+y=2\log \left(x\right)$

Convert it into $\frac{dy}{dx}+Py=Q$ form, where $P$ and $Q$ are functions of $x$.
$\Rightarrow \frac{dy}{dx}+\frac{1}{x\log \left(x\right)}y=\frac{2\log \left(x\right)}{x\log \left(x\right)}$
$\Rightarrow \frac{dy}{dx}+\frac{1}{x\log \left(x\right)}y=\frac{2}{x}$

Compare it with standard form, we get,
$P=\frac{1}{x\log \left(x\right)}$ and $Q=\frac{2}{x}$

Integrating factor is given as
$I.F.=e^{\int Pdx}$
Now,
$\int Pdx=\frac{1}{x\log \left(x\right)}dx$
Put $\log \left(x\right)=t$
$\Rightarrow \frac{dx}{x}=dt$
$\Rightarrow \int Pdx=\frac{dt}{t}=\log t=\log \left(\log \right(x\left)\right)$
$\therefore I.F.=e^{\log \left(\log \right(x\left)\right)}$
$\Rightarrow I.F.=\log \left(x\right)$