Question

I.F of $\frac{dy}{dx}+y=\frac{y+1}{x}$ is

• A. $\frac{x}{e^x}$
• B. $\frac{e^x}{x}$
• C. $x{e}^x$
• D. $e^x$

Answer 1

The correct option is B $\frac{e^x}{x}$

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

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

Here, $P\left(x\right)=1-\frac{1}{x}$ and $Q\left(x\right)=\frac{1}{x}$

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

$\Rightarrow I.F=e^{\int (1-\frac{1}{x})dx}$

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

$=e^x.e^{-\log x}$

$=e^x.e^{\log \frac{1}{x}}$

$\Rightarrow I.F=\frac{e^x}{x}$