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.
We have shown that the motion of a planet around the Sun, expressed in polar coordinates with the Sun at the origin, under general relativity follows the initial-value problem
,
,
,
where , , , is the gravitational constant of the universe, is the mass of the planet, is the mass of the Sun, is the constant angular momentum of the planet, is the speed of light, and is the smallest distance of the planet from the Sun during its orbit (i.e., at perihelion).
In this post, we will use the guesses
that arose from the technique/trick of reduction of order, where is some unknown function, to find the general solution of the differential equation
.
To do this, we will need to use the Product Rule for higher-order derivatives that was derived in the previous post:
and
.
In these formulas, Pascal’s triangle makes a somewhat surprising appearance; indeed, this pattern can be proven with mathematical induction.
We begin with . If , then
,
,
,
.
Substituting into the fourth-order differential equation, we find the differential equation becomes
The important observation is that the terms containing and cancelled each other. This new differential equation doesn’t look like much of an improvement over the original fourth-order differential equation, but we can make a key observation: if , then differentiating twice more trivially yields and . Said another way: if , then will be a solution of the original differential equation.
Integrating twice, we can find :
.
Therefore, a solution of the original differential equation will be
.
We now repeat the logic for :
.
Once again, a solution of this new differential equation will be , so that . Therefore, another solution of the original differential equation will be
.
Adding these provides the general solution of the differential equation:
.
Except for the order of the constants, this matches the solution that was presented earlier by using techniques taught in a proper course in differential equations.