Lecture Notes in Differential Equations - Bruce E. Shapiro

269
Thus
y = a 0
(
1 + 1 6 t3 + 1
+ a 1 t
)
12960 t9 + · · ·
180 t6 + 1
(
1 + 1
12 t3 + 1
504 t6 + 1
45360 t9 + · · ·
)
(28.109)
(28.110)
= a 0 y 1 (t) + a 1 y 2 (t) (28.111)
where y 1 and y 2 are defined by the sums in parenthesis. It is common to
define the Airy Functions
1
Ai(t) =
3 2/3 Γ(2/3) y 1
1(t) −
3 1/3 Γ(1/3) y 2(t) (28.112)
√ √
3
3
Bi(t) =
3 2/3 Γ(2/3) y 1(t) +
3 1/3 Γ(1/3) y 2(t) (28.113)
Either the sets {y 1 , y 2 } or {Ai, Bi} are fundamental sets of solutions to the
Airy equation.
Figure 28.1: Solutions to Airy’s equation. Left: The fundamental set y 1
(solid) and y 2 (dashed). Right: the traditional functions Ai(t)(solid) and
Bi(t) (dashed). The renormalization keeps Ai(t) bounded, whereas the unnormalized
solutions are both unbounded.
1
0.5
0
0.5
1
0.5
0.25
0.
0.25
0.5
15 10 5 0 5
15 10 5 0 5
Example 28.5. Legendre’s Equation of order n is given by
(1 − t 2 )y ′′ − 2ty ′ + n(n + 1)y = 0 (28.114)
where n is any integer. Equation (28.114) is actually a family of differential
equations for different values of n; we have already solved it for n = 2 in
example 28.3, where we found that most of the coefficients in the power
series solutions were zero, leaving us with a simple quadratic solution.