Question

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

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

Answer 1

Answer: (D)

$\frac{dy}{dx}=\frac{dy+y}{xy+x}$
$\Rightarrow \frac{dy}{dx}=\frac{y(x+1)}{x(y+1)}$
$\Rightarrow \int \left(\frac{y+1}{y}\right)dy=\int \left(\frac{x+1}{x}\right)dx$
$\Rightarrow \int \left(1+\frac{1}{y}\right)dy=\int \left(1+\frac{1}{x}\right)dx$
$\Rightarrow y+\log y=x+\log x+\log c$
$\Rightarrow y-x+\log y-\log x=\log c$
$\Rightarrow y-x+\log \frac{x}{y}=\log c$
$\Rightarrow y-x=\log c-\log \frac{y}{x}$
$\Rightarrow y-x=\log c+\log \frac{y}{x}$
$\Rightarrow y-x=\log \left(\frac{cx}{y}\right)$