# The general solution of the differential equation

Question:

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

A. $x y=C$

B. $x=C y^{2}$

C. $y=C x$

D. $y=C x^{2}$

Solution:

The given differential equation is:

$\frac{y d x-x d y}{y}=0$

$\Rightarrow \frac{y d x-x d y}{x y}=0$

$\Rightarrow \frac{1}{x} d x-\frac{1}{y} d y=0$

Integrating both sides, we get:

$\log |x|-\log |y|=\log k$

$\Rightarrow \log \left|\frac{x}{y}\right|=\log k$

$\Rightarrow \frac{x}{y}=k$

$\Rightarrow y=\frac{1}{k} x$

$\Rightarrow y=C x$ where $C=\frac{1}{k}$

Hence, the correct answer is C.