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 [[../DifferentialEquations/|differential equation]] $y^{\prime \prime }=0$ is called the differential equation of this family of straight lines because every [[../Bijective/|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 $n+1$ [[../Bijective/|relations]]. The differential equation that results is of order $n$.

References

[1] Lass, Harry. "Elements of pure and applied mathematics" New York: McGraw-Hill Companies, 1957.

This entry is a derivative of the Public [[../Bijective/|domain]] [[../Work/|work]] [1].