Question

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

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

Answer 1

The correct option is D

$\mathrm{y}-\mathrm{x}=\log \left(\frac{cx}{y}\right)$

Explanation of the correct option.

Compute the required value.

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

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

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

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

Now, Integrate both sides.

$\Rightarrow$ $\log \left(y\right)+y=\log \left(x\right)+x+\log \left(C\right)$

$\Rightarrow$ $\log \left(\frac{y}{x}\right)=x-y+\log \left(C\right)$

$\Rightarrow$ $y-x=\log \left(C\right)-\log \left(\frac{y}{x}\right)$

$\Rightarrow$ $\mathrm{y}-\mathrm{x}=\log \left(\frac{cx}{y}\right)$

Hence option $\left(D\right)$ is the correct option.