Lecture Notes in Differential Equations - Bruce E. Shapiro

Lesson 30
The Method of Frobenius
If the equation
y ′′ + b(t)y ′ + c(t)y = 0 (30.1)
has a regular singularity at the point t = t 0 , then the functions
p(t) = (t − t 0 )b(t) (30.2)
q(t) = (t − t 0 ) 2 c(t) (30.3)
are analytic at t = t 0 (see the discussion following (28.65)). Substituting
(30.2) and (30.3) into (30.1) and then multiplying the result through by
(t − t 0 ) 2 gives
(t − t 0 ) 2 y ′′ + (t − t 0 )p(t)y ′ + q(t)y = 0 (30.4)
Thus we are free to take equation (30.4) as the canonical form any second
order differential equation with a regular singularity at t = t 0 . By canonical
form, we mean a standard form for describing any the properties of any
second order differential equation with a regular singularity at t = t 0 .
The simplest possible form that equation (30.4) can take occurs when t 0 = 0
and p(t) = q(t) = 1,
t 2 y ′′ + ty ′ + y = 0 (30.5)
Equation (30.5) is a form of the Cauchy-Euler equation
t 2 y ′′ + aty ′ + by = 0, (30.6)
where a and b are nonzero constants. The fact that theorem 28.1 fails
to guarantee a series solution around t = 0 is illustrated by the following
example.
283