Solving Differential Equations — the Variable-Separable Method

When variables can be separated

A first-order differential equation is variable-separable if it can be rearranged so that ALL the $y$-terms (including $dy$) are on one side, and ALL the $x$-terms (including $dx$) are on the other:

Integrating both sides independently, and combining the two arbitrary constants into one, gives the general solution.

Worked reasoning

For $\frac{dy}{dx}=\frac{y}{x}$, separate variables: $\frac{dy}{y}=\frac{dx}{x}$. Integrating both sides:

Exponentiating both sides (and writing $k=e^C$ as a new arbitrary constant) gives the general solution $y=kx$.