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

y'' + p(x) y' + q(x) y = 0

and reduced it to

u'' + r(x) u = 0

using the change of variable

u(x) = \exp\left( \frac{1}{2} \int^x p(t)\,dt\right ) y(x)

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

p(x) = ax^n + bx^{n-1} + cx^{n-2} + \cdots

and use the change of variables

x = t - \frac{b}{na}

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

P(D) y = 0

We can turn this into a differential

Q(D) u = 0

where the polynomial

Q(D) = P\left(D - \frac{p}{n}\right)

has no term involving Dn−1 by solving

\left(D - \frac{p}{n}\right) u = D y

which leads to

u(x) = \exp\left( \frac{1}{n} \int^x p(t)\,dt\right ) y(x)

generalizing the result above for second order ODEs.