Question

Find the general solution of the differential equation $\frac{dy}{dx}-\frac{y}{x}=0$.

Answer 1

$\frac{dy}{dx}-\frac{y}{x}=0$

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

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

Integrating both sides, $\log y=\log x+c$ where c is an arbitrary constant.

$\Rightarrow \log y=\log kx$, $k$ being greater than zero.

$\Rightarrow y=kx$ is the general solution