Question

Solve the differential equation $\frac{dy}{dx}=\frac{xy+y}{xy+x}$.

Answer 1

We need to solve $\frac{dy}{dx}=\frac{xy+y}{xy+x}$

Consider $\frac{dy}{dx}=\frac{xy+y}{xy+x}$

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

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

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

Integrating both sides we get

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

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

$\Rightarrow y-x=\ln x-\ln y+\ln c$

$\Rightarrow y-x=\ln (c\cdot \left(\frac{x}{y}\right))$