Question

I.F of $x\log x\frac{dy}{dx}+y=2\log x$ is

• A. $x$
• B. $\log x$
• C. $\log \left(\log x\right)$
• D. $\frac{2}{x}$

Answer 1

The correct option is B $\log x$

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

Dividing the above equation by $x\log x$, we get

$\frac{dy}{dx}+\frac{1}{x\log x}y=\frac{2}{x}$

We get $P\left(x\right)=\frac{1}{x\log x}$ and $Q\left(x\right)=\frac{2}{x}$

$I.F=e^{\int P\left(x\right)dx}$

$=e^{\int \frac{1}{x\log x}dx}$

Take $\log x=t\Rightarrow \frac{1}{x}dx=dt$

$\Rightarrow I.F=e^{\int \frac{dt}{t}}=e^{\log t}=t=\log x$

$\Rightarrow I.F=\log x$