PlanetPhysics/Differential Equation of a Family of Curves

The family of straight lines in the plane is characterized by the equation $$y = ax + b$$, where $$a$$ and $$b$$ are arbitrary constants. To fix these constants means to select one member of the family, that is, to fix our attention on one straight line among all the others. The differential equation $$y^{\prime \prime} = 0$$ is called the differential equation of this family of straight lines because every function of the form $$y = ax +b$$ satisfies this equation and, conversely, every solution of $$y^{\prime \prime} = 0$$ is a member of the family $$y = ax + b$$. The differential equation $$y^{\prime \prime}=0$$ characterizes the family as a whole without specific reference to the particular members.

More generally, a family of curves can be described by

$$ y = f(x,a_1,a_2,\dots,a_n)$$

or implicitly by $$F(x,y,a_1,a_2,\dots,a_n) = 0$$, in which $$n$$ arbitrary constants appear. The differential equation of the family is obtained by successively differentiating $$n$$ times and eliminating the constants between the resulting $$n + 1$$ relations. The differential equation that results is of order $$n$$.