Lecture Notes in Differential Equations - Bruce E. Shapiro

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Table 28.1: Table of Special Functions defined by Differential Equations.
Name
Differential Equation
of Equation Names of Solutions
Airy
y ′′ = ty
Equation Airy Functions Ai(t), Bi(t)
Bessel t 2 y ′′ + ty ′ + (t 2 − ν 2 )y = 0, 2ν ∈ Z +
Equation Bessel Functions J ν (t)
Neumann Functions Y ν (t)
Modified t 2 y ′′ + ty ′ − (t 2 + ν 2 )y = 0, ν ∈ Z +
Bessel
Modified Bessel Functions I ν (t), K ν (x)
Equation Hankel Functions H ν (t)
Euler
t 2 y ′′ + αty ′ + βy = 0, α, β ∈ C
Equation t r , where r(r − 1) + αr + β = 0
Hermite y ′′ − 2ty ′ + 2ny = 0, n ∈ Z +
Equation Hermite polynomials H n (t)
Hypergeometric t(1 − t)y ′′ + (c − (a + b + 1)t)y ′ − aby = 0, a, b, c, d ∈ R
Equation Hypergeometric Functions F , 2 F 1
Jacobi
t(1 − t)y ′′ + [q − (p + 1)t]y ′ + n(p + n)y = 0, n ∈ Z, a, b ∈ R
Equation Jacobi Polynomicals J n
Kummer ty ′′ + (b − t)y ′ − ay = 0, a, b ∈ R
Equation Confluent Hypergeometric Functions 1 F 1
Laguerre ty ′′ + (1 − t)y ′ + my = 0, m ∈ Z +
Equation Laguerre Polynomials L m (t)
Associated ty ′′ + (k + 1 − t)y
Laguerre Eqn. Associated Laguerre Polynomials L k m(t)
Legendre (1 − t 2 )y ′′ − 2ty ′ + n(n + 1)y = 0, n ∈ Z +
Equation Legendre Polynomials
[
P n (t)
Associated (1 − t 2 )y ′′ − 2ty ′ + n(n + 1) − m2 y = 0, m, n ∈ Z +
1−t 2 ]
Legendre Eq. Associate Legendre Polynomials Pn m (t)
Tchebysheff (1 − t 2 )y ′′ − ty ′ + n 2 y = 0 (Type I)
Equation (1 − t 2 )y ′′ − 3ty ′ + n(n + 2)y = 0 (Type II)
Tchebysheff Polynomials T n (t), U n (t)