Question

The solution of the differential equation $x\log x\frac{dy}{dx}+y=2\log x$ is:

• A. $y=\log x+\frac{c}{\log x}$
• B. $y=\log x-\frac{c}{\log x}$
• C. $y=\log x-c\log x$
• D. none of these

Answer 1

Answer: (A)

$y=\log x+\frac{c}{\log x}$

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

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

$I.F.=e^{\int \frac{1}{x\log x}dx}=e^{\log \left(\log x\right)}=\log x$

Hence, the solution of the differential equation is

$y.\log x=\int \frac{2}{x}\log xdx+c$

$\Rightarrow y.\log x={\left(\log x\right)}^2+c\Rightarrow y=\log x+\frac{x}{\log x}$