This post ties together two earlier posts: the previous post on a change of variable to remove a term from a polynomial, and an older post on a change of variable to remove a term from a differential equation. These are different applications of the same idea.
A linear differential equation can be viewed as a polynomial in the differential operator D applied to the function we’re solving for. More on this idea here. So it makes sense that a technique analogous to the technique used for “depressing” a polynomial could work similarly for differential equations.
In the differential equation post mentioned above, we started with the equation
and reduced it to
using the change of variable
So where did this change of variables come from? How might we generalize it to higher-order differential equations?
In the post on depressing a polynomial, we started with a polynomial
and use the change of variables
to eliminate the xn-1 term. Let’s do something analogous for differential equations.
Let P be an nth degree polynomial and consider the differential equation
We can turn this into a differential
where the polynomial
has no term involving Dn-1 by solving
which leads to
generalizing the result above for second order ODEs.