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:
 * $$g \left({x, y, \frac{\mathrm{d}{y}}{\mathrm{d}{x}}, c}\right) = 0$$

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

which is a first order ordinary differential equation.