## ABSTRACT

An Euler equation is a homogeneous linear differential equation which can be written so that each term is a constant × x^{m} y
^{(m)}. For example, a second-order Euler equation is of the form
a
x
2
y
″
+
b
x
y
′
+
c
y
=
0
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where a, b and c are constants with a ≠ 0. These equations are important for several reasons:

They are easily solved.

They occasionally arise in applications, though not nearly as often as equations with constant coefficients.

They are the simplest examples of a broad class of differential equations for which infinite series solutions can be obtained using the “method of Frobenius”.

In this chapter it is seen that simple modifications of the methods developed for constant coefficient equations apply towards solving Euler equations. In particular, assuming y = x^{r}
leads to a polynomial equation (the “indicial equation”) for r, and a fundamental set of solutions to the differential equation can then be constructed from the roots of this polynomial equation (provided the multiplicities of those roots are also known). The chapter ends with an analysis of a direct link between Euler equations and homogeneous linear differential equations with constant coefficients.