Lecture Notes in Differential Equations - Bruce E. Shapiro

286 LESSON 30. THE METHOD OF FROBENIUS
Since each term in (30.20) has the form t α for some complex number α,
this led Georg Ferdinand Frobenius to look for solutions of the form
y = (t − t 0 ) α S(t) (30.32)
where S(t) is analytic at t 0 and can be expanded in a power series,
∞∑
S(t) = c k (t − t 0 ) k (30.33)
k=0
To determine the condition under which Frobenius’ solution works, we differentiate
(30.32) twice
y ′ = α(t − t 0 ) α−1 S + (t − t 0 ) α S ′ (30.34)
y ′′ = α(α − 1)(t − t 0 ) α−2 S + 2α(t − t 0 ) α−1 S ′ + (t − t 0 ) α S ′′ (30.35)
and substitute equations (30.32), (30.34), and (30.35) into the differential
equation (30.4):
0=(t − t 0 ) 2 [ α(α − 1)(t − t 0 ) α−2 S + 2α(t − t 0 ) α−1 S ′ + (t − t 0 ) α S ′′]
+(t − t 0 ) [ α(t − t 0 ) α−1 S + (t − t 0 ) α S ′] p(t) + (t − t 0 ) α Sq(t) (30.36)
=(t − t 0 ) α+2 S ′′ + [2α + p(t)] (t − t 0 ) α+1 S ′
+ [α(α − 1) + αp(t) + q(t)] (t − t 0 ) α S (30.37)
For t ≠ t 0 , the common factor of (t − t 0 ) α can be factored out to and the
result becomes
0 = (t−t 0 ) 2 S ′′ +[2α + p(t)] (t−t 0 )S ′ +[α(α − 1) + αp(t) + q(t)] S (30.38)
Since p(t), q(t), and S(t) are all analytic at t 0 the expression on the righthand
side of (30.38) is also analytic, and hence infinitely differentiable, at
t 0 . Thus it must be continuous at t 0 (differentiability implies continuity),
and its limit as t → t 0 must equal its value at t = t 0 . Thus
[α(α − 1) + αp(t 0 ) + q(t 0 )] S(t 0 ) = 0 (30.39)
So either S(t 0 ) = 0 (which means that y(t 0 ) = 0) or
α(α − 1) + αp(t 0 ) + q(t 0 ) = 0 (30.40)
Equation (30.40) is called the indicial equation of the differential equation
(30.4); if it is not satisfied, the Frobenius solution (30.32) will not work.
The Indicial equation plays a role analogous to the characteristic equation
but at regular singular points.
Before we prove that the indicial equation is also sufficient, we present the
following example that demonstrates the Method of Frobenius.