Question

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

• A. $x+y-\log \frac{cy}{x}$
• B. $x+y=\log \left(cxy\right)$
• C. $x-y-\log \frac{cx}{y}$
• D. $y-x=\log \frac{cx}{y}$

Answer 1

Answer: (D)

$y-x=\log \frac{cx}{y}$

Given differential equation is
$\frac{dy}{dx}=\frac{xy+y}{xy+x}$
$\Rightarrow \frac{dy}{dx}=\frac{y(1+x)}{x(1+y)}$
$\Rightarrow \left(\frac{1+y}{y}\right)dy=\left(\frac{1+x}{x}\right)dx$

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

$\Rightarrow \log y+y=\log x+x+c$

$\Rightarrow y-x=\log \frac{cx}{y}$