Lecture Notes in Differential Equations - Bruce E. Shapiro

278 LESSON 29. REGULAR SINGULARITIES
for some unknown number (integer, real, or possibly complex) r. To see
what values of r might work, we substitute into the original ODE (29.4):
t 2 r(r − 1)t r−2 + αtrt r−1 + βt r = 0 (29.16)
=⇒ t r [r(r − 1) + αr + β] = 0 (29.17)
Since this must hold for all values of t, the second factor must be identically
equal to zero. Thus we obtain
r(r − 1) + αr + β = 0 (29.18)
without the restriction that r ∈ Z. We call this the indicial equation
(even though there are no indices) due to its relationship (and similarity)
to the indicial equation that we will have to solve in Frobenius’ method in
chapter 30.
We will now consider each of the three possible cases for the roots
r 1,2 = 1 − α ± √ (1 − α) 2 − 4β
2
(29.19)
Case 1: Two real roots. If r 1 , r 2 ∈ R and r 1 ≠ r 2 , then we have found two
linearly independent solutions and the general solution is
Example 29.4. Find the general solution of
In standard from this becomes
The indicial equation is
y = C 1 t r1 + C 2 t r2 (29.20)
2t 2 y ′′ + 3ty ′ − y = 0 (29.21)
t 2 y ′′ + 3 2 ty′ − 1 2 y = 0 (29.22)
0 = r(r − 1) − 3 2 r + 1 2 = 2 2 r2 + 1 2 r + 1 2
(29.23)
0 = 2r 2 + r + 1 = (2r − 1)(r + 1) =⇒ r = 1 , −1 (29.24)
2
Hence
y = C 1
√
t +
C 2
t
(29.25)