Overview
There is sometimes a need to elimate arbitrary constants from an equation, and the best way to do this is by use of the calculus. You'll need a basic knowledge of that discipline to make this write-up worthwhile.
Eliminating Arbitrary Constants
Consider this very simple example:
Example 1: y = x^2 + px
p is some arbitrary constant, that we would now like to eliminate. Differentiating:
dy/dx = 2x + p
p = dy/dx - 2x
Substituting this value for p into the original equation:
y = x2 + (dy/dx - 2x)x
There is no need to multiply this out, as p has evidently been removed.
Generally elimation of one constant is straight-forward, and removing multiple constants requires a little lateral thought. For example, the equation of simple harmonic motion:
Example 2: x = A*cos(pt - a)
Differentiate to give:
dx/dt = -pA*sin(pt - a)
and again...
d2x/dt2 = -p2A*cos(pt - a) = -p2x
In two steps, A and a have been eliminated. This method is easily extended to other examples. The basic technique is that, to eliminate n constants, it is necessary to differentiate n times to create n+1 equations, and then solve them just as with any set of simultaneous equations.
Practice Examples
If you need practice for consolidation, attempt to eliminate the arbitrary constant from these:
1. y = Px + P^3
2. y = A*eBx
Credit For This Write-Up
Some examples adapted, others lifted directly, from the classic Differential Equations, H.T.H. Piaggio, 1920.
Solutions
1. y = (dy/dx)x + (dy/dx)3
2. y * d2y/dx2 = (dy/dx)2