Question

The general solution of the differential equation $\frac{ydx-xdy}{y}=0$ is

a) $xy=C$

b) $x=C{y}^2$

c) $y=Cx$

d) $y=C{x}^2$

Answer 1

Given, differential equation is $\frac{ydx-xdy}{y}=0$
$\Rightarrow ydx-xdy=0\Rightarrow \frac{1}{x}dx-\frac{1}{y}dy=0$
On integrating both sides, we get
$log|x|-log|y|=logk\Rightarrow log\left|\frac{x}{y}\right|=logk\Rightarrow \frac{x}{y}=k$
$\Rightarrow y=\frac{1}{k}x\Rightarrow y=Cx$ $(let\frac{1}{k}=C)$
Hence, the correct option is (C).