In this series, I’m discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury’s orbit and thus confirm Einstein’s theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago.
In the last few posts, I’ve used a standard technique from differential equations: to solve the th order homogeneous differential equation with constant coefficients
,
we first solve the characteristic equation
using techniques from Precalculus. The form of the roots determines the solutions of the differential equation.
While this is a standard technique from differential equations, the perspective I’m taking in this series is scaffolding the techniques used to predict the precession in a planet’s orbit using only techniques from Calculus and Precalculus. So let me discuss why the above technique works, assuming that the characteristic equation does not have repeated roots. (The repeated roots case is a little more complicated but is not needed for the present series of posts.)
We begin by guessing that the above differential equation has a solution of the form . Differentiating, we find , , etc. Therefore, the differential equation becomes
The last step does not “lose” any possible solutions for since can never be equal to . Therefore, solving the differential equation reduces to finding the roots of this polynomial, which can be done using standard techniques from Precalculus.
For example, one of the differential equations that we’ve encountered is . The characteristic equation is , which has roots . Therefore, two solutions to the differential equation are and , so that the general solution is
.
To write this in a more conventional way, we use Euler’s formula , so that
.
Likewise, in the previous post, we encountered the fourth-order differential equation . To find the roots of the characteristic equation, we factor:
.
Therefore, four solutions of this differential equation are , , , and , so that the general solution is
.
Using Euler’s formula as before, this can be rewritten as
.
Published by John Quintanilla
I’m a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT’s program for preparing secondary teachers of mathematics and science.
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Published
May 24, 2024February 13, 2024
