Question

Find the integrating factor for the following differential equation: $x\log x\frac{dy}{dx}+y=2\log x$.

Answer 1

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

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

Let $logx=t\Rightarrow \frac{1}{x}dx=dt$

$\Rightarrow$integrating factor $=e^{\int \frac{dt}{t}}$

$\Rightarrow e^{lnt}=e^{ln\left(lnx\right)}$

$\Rightarrow$integrating factor$=e^{ln\left(lnx\right)}$