Question

Solution of the differential equation
$dx(\frac{1}{y}-\frac{1}{x})+dy(\frac{1}{y}-\frac{x}{y^2})=0$ is

• A. $In\left(\frac{y}{x}\right)+\frac{x}{y}=K$
• B. $In\left(\frac{x^2}{y^2}\right)=K$
• C. $In\left(\frac{y}{x}\right)+\frac{y}{x}=K$
• D. $2In\left(\frac{y}{x}\right)+\frac{x}{y}=K$

Answer 1

Answer: (B)

$In\left(\frac{x^2}{y^2}\right)=K$

$dx\left(\frac{1}{y}-\frac{1}{x}\right)+dy\left(\frac{1}{y}-\frac{x}{y^2}\right)=0$

$\Rightarrow \left(\frac{x-y}{xy}\right)dx=-\left(\frac{y-x}{y^2}\right)dy$

$\Rightarrow \left(\frac{x-y}{xy}\right)dx=\left(\frac{x-y}{y^2}\right)dy$

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

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

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

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

$\Rightarrow 2\ln \left(\frac{x}{y}\right)=2\ln c$

$\Rightarrow \ln \left(\frac{x^2}{y^2}\right)=K$