Lecture Notes in Differential Equations - Bruce E. Shapiro

416 APPENDIX C. SUMMARY OF METHODS
To solve
a(t)y ′′ + b(t)y ′ + c(t)y = g(t)
about a point t 0 where a(t) = 0 but the limits b(t)/a(t) and c(t)/a(t) exist
as t → 0 (a regular singularity), solve the indicial equation
r(r − 1) + rp 0 + q 0 = 0
for r where p 0 = lim t→0 b(t 0 )/a(t 0 ) and and q 0 = lim t→0 c(t 0 )/a(t 0 ). Then
one solution to the homogeneous equation is
y(t) = (t − t 0 ) r
∞
∑
k=0
c k (t − t 0 ) k
for some unknown coefficients c k .Determine the coefficients by linear independance
of the powers of t. The second solution is found by reduction of
order and the particular solution by variation of parameters.
Method of Frobenius
To solve
(t − t 0 ) 2 y ′′ + (t − t 0 )p(t)y ′ + q(t)y = 0
where p and q are analytic at t 0 , let p 0 = p(0) and q 0 = q(0) and find the
roots α 1 ≥ α 2 of
α 2 + (p 0 − 1)α + q 0 = 0
Define ∆ = α 1 − α 2 . Then for some unknowns c k , a first solution is
y 1 (t) = (t − t 0 ) α1
∞ ∑
k=0
c k (t − t 0 ) k
If ∆ ∈ R is not an integer or the roots are complex,
y 2 (t) = (t − t 0 ) α2
∞ ∑
k=0
a k (t − t 0 ) k
If α 1 = α 2 = α, then y 2 = ay 1 (t) ln |t − t 0 | + (t − t 0 ) α
If ∆ ∈ Z, then y 2 = ay 1 (t) ln |t − t 0 | + (t − t 0 ) α2
∞
∑
k=0
∞
∑
k=0
a k (t − t 0 ) k
a k (t − t 0 ) k