Family of Solutions

A family of solutions to a differential equation is a collection of solution curves obtained by introducing one or more free constants (parameters) into the general solution.

Formally, a one-parameter family can be written as

$$\varphi (x,y,C)=0\text{or}y=f(x,C),C\in \mathbb{R}$$

where each fixed value of $C$ gives a distinct solution of the differential equation.

Geometrically, a family of solutions is a smooth collection of curves filling a region of the plane.