**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 = x^{2} + (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...

d^{2}x/dt^{2} = -p^{2}A*cos(pt - a) = -p^{2}x

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*e^{Bx}

**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 * d^{2}y/dx^{2} = (dy/dx)^{2}