Answer:

7.6: The Method of Frobenius I

Last updatedAug 13, 2020

7.5E: Regular SINGULAR Points Euler Equations (Exercises)

7.6E: The Method of Frobenius I (Exercises)

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Contributed by William F. Trench

Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University

Sections 7.5-7.7 deal with three distinct cases satisfying the assumptions introduced in Section 7.4. In all three cases, (A) has at least ONE solution of the form

y1=xr

∞

∑

n=0 anxn,

where r need not be an integer. The problem is that there are three possibilities - each requiring a different approach - for the form of a second solution y2 such that {y1,y2} is a fundamental pair of solutions of (A).

In this section we begin to study series solutions of a homogeneous LINEAR second order differential equation with a regular singular point at x0=0, so it can be written as

x2A(x)y″+xB(x)y′+C(x)y=0,

where A, B, C are polynomials and A(0)≠0.

We’ll see that Equation 7.6.1 always has at least one solution of the form

y=xr

∞

∑

n=0 anxn