Eliminating terms from higher-order differential equations

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 x^n-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 D^n-1 by solving

which leads to

generalizing the result above for second order ODEs.

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