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:
 * $y = \map y {x, c}$

where different values of $c$ give different curves.

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

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

Differentiate $f$ $x$ to get a relation in the form:
 * $\map g {x, y, \dfrac {\d y} {\d x}, c} = 0$

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

which is a first order ordinary differential equation.