An indicial equation, also called a characteristic equation, is a recurrence equation obtained during application of the Frobenius method of solving a second-order ordinary differential equation. The indicial equation is obtained by noting that, by definition, the lowest order term  (that corresponding to ) must have a coefficient of zero.

1. If the two roots are equal, only one solution can be obtained.

2. If the two roots differ by a noninteger, two solutions can be obtained.

3. If the two roots differ by an integer, the larger will yield a solution. The smaller may or may not.

For an example of the construction of an indicial equation, see Bessel function of the first kind.

The following table gives the indicial equations for some common differential equations.

differential equation	indicial equation
Bessel differential equation
Chebyshev differential equation
Hermite differential equation
Jacobi differential equation
Laguerre differential equation
Legendre differential equation

REFERENCES:

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 532-534, 1953.