Question

Solution of differential equation $xdy-ydx=0$ is

Answer 1

Given: $xdy-ydx=0$
$\Rightarrow \frac{dy}{y}=\frac{dx}{x}$
Integrating on both sides, we get
$\Rightarrow \int \frac{dy}{y}=\int \frac{dx}{x}\Rightarrow \ln |y|=\ln |x|+c\Rightarrow \ln |y|-\ln |x|=c\Rightarrow \ln \left|\frac{y}{x}\right|=c$
$\Rightarrow \left|\frac{y}{x}\right|=e^c\Rightarrow \frac{y}{x}=\pm e^c=k\Rightarrow y=kx$
Thus, $y=kx$ represent a straight line passing through origin.