One-Parameter Family of Curves for First Order ODE

Theorem
Every one-parameter family of curves is the general solution of some first order ordinary differential equation.

Conversely, every first order ordinary differential equation has as its general solution some one-parameter family of curves.

Proof
From Picard's Existence Theorem, every point in a given rectangle is passed through by some curve which is the solution of a given integral curve of a differential equation.

The equation of this family can be written as:
 * $y = y \left({x, c}\right)$

where different values of $c$ give different curves.

The integral curve which passes through $\left({x_0, y_0}\right)$ corresponds to the value of $c$ such that:
 * $y_0 = y \left({x_0, c}\right)$

Conversely, consider the one-parameter family of curves described by:
 * $f \left({x, y, c}\right) = 0$

Differentiate $f$ WRT $x$ to get a relation in the form:
 * $g \left({x, y, \dfrac{\mathrm{d}{y}}{\mathrm{d}{x}}, c}\right) = 0$

Then eliminate $c$ between these expressions for $f$ and $g$ and get:
 * $F \left({x, y, \dfrac{\mathrm{d}{y}}{\mathrm{d}{x}}}\right) = 0$

which is a first order ordinary differential equation.