Question

The solution to the differential equation $\frac{ dy }{ dx } \ln x +\frac{ y }{ x }=\frac{ dy }{ dx }$ which passes through the point $\left( e ^2, 3\right)$ is

• A. $y=3 e^x-\left(3 e^{e^2}-3\right)$
• B. $y =3 \ln x -3$
• C. $y =\frac{3}{\ln x -1}$
• D. $y=\frac{9}{1+\ln x}$

Answer 1

Answer: (C)

$ \frac{ dy }{ dx } \ln x +\frac{ y }{ x }=\frac{ dy }{ dx }$
$\frac{ d }{ dx }( y \ln x )=\frac{ dy }{ dx }$
$\text { Integrating }$
$y \ln x = y + C \text { Pass }\left( e ^2, 3\right)$
$3 \cdot 2=3+ C \Rightarrow C =3 $
$y \ln x = y +3 $
$ y =\frac{3}{\ln x -1} \cdot $