Question

The general solution of the differential equation $x\frac{dy}{dx}+xy=e^{-x}$ is

• A. $y\cdot e^{-x}=\ln |x|+C$
• B. $y\cdot e^x=\ln \frac{1}{|x|}+C$
• C. $y\cdot e^x=\ln |x|+C$
• D. $y\cdot e^x=C$

Answer 1

The correct option is C $y\cdot e^x=\ln |x|+C$

$x\frac{dy}{dx}+xy=e^{-x}\Rightarrow \frac{dy}{dx}+y=\frac{e^{-x}}{x}$
Integrating factor
$=e^{\int 1dx}=e^x$
Solution of the D.E,
$y\cdot e^x=\int e^x\times \frac{e^{-x}}{x}dx+C\Rightarrow y\cdot e^x=\int \frac{1}{x}dx+C\Rightarrow y\cdot e^x=\ln |x|+C$