We consider the utilization of power series to determine solutions to differential equations near a singular point. We also study Euler equations.

Regular Singular Points: Euler Equations

In the next three sections we’ll continue to study equations of the form

where , , and are polynomials, but the emphasis will be different from that of Trench 7.2 and 7.3, where we obtained solutions of (eq:7.4.1) near an ordinary point in the form of power series in . If is a singular point of (eq:7.4.1) (that is, if ), the solutions can’t in general be represented by power series in . Nevertheless, it’s often necessary in physical applications to study the behavior of solutions of (eq:7.4.1) near a singular point. Although this can be difficult in the absence of some sort of assumption on the nature of the singular point, equations that satisfy the requirements of the next definition can be solved by series methods discussed in the next three sections. Fortunately, many equations arising in applications satisfy these requirements.

For convenience we restrict our attention to the case where is a regular singular point of (eq:7.4.2). This isn’t really a restriction, since if is a regular singular point of (eq:7.4.2) then introducing the new independent variable and the new unknown leads to a differential equation with polynomial coefficients that has a regular singular point at .

Euler Equations

The simplest kind of equation with a regular singular point at is the Euler equation, defined as follows.

Theorem thmtype:5.1.1 implies that (eq:7.4.6) has solutions defined on and , since (eq:7.4.6) can be rewritten as For convenience we’ll restrict our attention to the interval . The key to finding solutions on is that if then is defined as a real-valued function on for all values of , and substituting into (eq:7.4.6) produces

The polynomial is called the indicial polynomial of (eq:7.4.6), and is its indicial equation. From (eq:7.4.7) we can see that is a solution of (eq:7.4.6) on if and only if . Therefore, if the indicial equation has distinct real roots and then and form a fundamental set of solutions of (eq:7.4.6) on , since is nonconstant. In this case is the general solution of (eq:7.4.6) on .

If the indicial equation has a repeated root , then is a solution of

on , but (eq:7.4.10) has no other solution of the form . If the indicial equation has complex conjugate zeros then (eq:7.4.10) has no real–valued solutions of the form . Fortunately we can use the results of Trench 5.2 for constant coefficient equations to solve (eq:7.4.10) in any case.

We first show that satisfies (eq:7.4.12) on if and only if satisfies the constant coefficient equation on . To do this, it’s convenient to write , or, equivalently, ; thus, , where . From the chain rule, and, since it follows that Differentiating this with respect to and using the chain rule again yields
From this and (eq:7.4.14), Substituting this and (eq:7.4.14) into (eq:7.4.12) yields (eq:7.4.13). Since (eq:7.4.11) is the characteristic equation of (eq:7.4.13), Theorem thmtype:5.2.1 implies that the general solution of (eq:7.4.13) on is
Since , substituting in the last three equations shows that the general solution of (eq:7.4.12) on has the form stated in the theorem.

Text Source

Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)