Degenerate hypergeometric equation.. The results of solving the original equation are presented in Table T5.1... The Bessel functions are described in Section 18.6 in detail.. Modified B
Trang 15. y
xx + (ax + b)y x + (αx2+ βx + γ)y = 0.
The substitution y = u exp(sx2), where s is a root of the quadratic equation 4s2+2as +α =0,
leads to an equation of the form T5.2.1.11: u xx +[(a+4s)x+b]u x +[(β +2bs)x+γ +2s]u =0
6. xy
xx + ay x + by = 0.
1◦ The solution is expressed in terms of Bessel functions and modified Bessel functions:
y=
⎧
⎨
⎩
x1 –a
2
C1J ν 2√ bx
+ C2Y ν 2√ bx
if bx >0,
x1 –a
2
C1I ν 2√|bx| + C2K ν 2√|bx| if bx <0,
where ν =|1– a|
2◦ For a = 1
2(2n+1), where n =0, 1, , the solution is
y=
⎧
⎪
⎪
C1 d
n
dx n cos
√
4bx + C2 d n
dx n sin
√
4bx if bx >0,
C1 d
n
dx n cosh
4|bx|+ C2 d n
dx n sinh
4|bx| if bx <0
7. xy
xx + ay x + bxy = 0.
1◦ The solution is expressed in terms of Bessel functions and modified Bessel functions:
y=
⎧
⎨
⎩
x1 –a
2
C1J ν b x
+ C2Y ν b x
if b >0,
x1 –a
2
C1I ν |b|x
+ C2K ν |b|x
if b <0,
where ν = 12|1– a|
2◦ For a =2n, where n =1, 2, , the solution is
y=
⎧
⎪
⎪
C1
x
d dx
n
cos x √
b
+ C2
x
d dx
n
sin x √
b
if b >0,
C1
x
d dx
n
cosh x √
–b
+ C21
x
d dx
n
sinh x √
–b
if b <0
8. xy
xx + ny x + bx1–2n y = 0.
For n =1, this is the Euler equation T5.2.1.12 For n≠ 1, the solution is
y=
⎧
⎪
⎪
⎪
⎩
C1sin
√
b
n–1x1–n
+ C2cos
√
b
n–1x1–n
if b >0,
C1exp
√
–b
n–1x1–n
+ C2exp
–√ –b
n–1x1–n
if b <0
9. xy
xx + ay x + bx n y = 0.
If n = –1and b =0, we have the Euler equation T5.2.1.12 If n≠–1and b≠ 0, the solution
is expressed in terms of Bessel functions:
y = x1–2a
C1J ν
n+1x
n+1
2
+ C2Y ν
n+1x
n+1
2
, where ν = |1– a|
n+1.
Trang 210. xy
xx + (b – x)y x – ay = 0.
Degenerate hypergeometric equation.
1◦ If b≠ 0, –1, –2, –3, , Kummer’s series is a particular solution:
∞
k=1
(a) k (b) k
x k
k!, where (a) k = a(a +1) (a + k –1), (a)0=1 If b > a >0, this solution can be written in terms of a definite integral:
Φ(a, b; x) = Γ(a) Γ(b – a) Γ(b) 01e xt t a–1(1– t) b–a–1dt,
where Γ(z) = ∞
–t t z–1dtis the gamma function.
If b is not an integer, then the general solution has the form:
y = C1Φ(a, b; x) + C2x1 –b Φ(a – b +1, 2– b; x).
2◦ For b≠ 0, –1, –2, –3, , the general solution of the degenerate hypergeometric equation
can be written in the form
y = C1Φ(a, b; x) + C2Ψ(a, b; x),
while for b =0, –1, –2, –3, , it can be represented as
y = x1–b
C1Φ(a – b +1, 2– b; x) + C2Ψ(a – b +1, 2– b; x)
The functionsΦ(a, b; x) and Ψ(a, b; x) are described in Subsection 18.9 in detail.
11. (a2x + b2)y xx + (a1x + b1)y x + (a0x + b0)y = 0.
Let the functionJ (a, b; x) be an arbitrary solution of the degenerate hypergeometric equa-tion xy xx + (b – x)y x – ay =0 (see T5.2.1.10), and let the function Z ν (x) be an arbitrary solution of the Bessel equation x2y
xx + xy x + (x2– ν2)y =0 (see T5.2.1.13) The results
of solving the original equation are presented in Table T5.1
TABLE T5.1 Solutions of equation T5.2.1.11 for different values of the determining parameters
Solution: y = e kx w (z), where z = x – μ
λ
a2≠ 0 ,
a2≠ 4a0a2
√
D – a1
2a2 – a2
2a2k + a1 –b2
a2 J (a, b; z) a = B(k)/(2a2k + a1),
b = (a2b1– a1b2)a–2
a2 = 0 ,
a1≠ 0 –
a0
a1 1 – 2b2k + b1
a1 J a, 12; βz2 a = B(k)/(2a1),
β = –a1/( 2b2)
a2 ≠ 0 ,
a2= 4a0a2 – 2a a12 a2 –b2
a2 z
ν/2
Z ν β √
z ν= 1 – ( 2b2k + b1)a–1,
β= 2 B (k)
a2= a1 = 0 ,
a0≠ 0 –
b1
2b2 1 b2–4b0b2
4a0b2
z1/2Z1/3 βz3/2
; see also 2.1.2.12 β=
2 3
a
0
b2
1/2
Notation: D = a2– 4a0a2, B(k) = b2k2+ b1k + b0
Trang 312. x2y
xx + axy x + by = 0.
Euler equation Solution:
y=
⎧
⎪
⎪
⎪
⎪
|x|1–2a C1|x|μ + C2|x|–μ
if (1– a)2>4b,
|x|1–2a (C1+ C2ln|x|) if (1– a)2=4b,
|x|1–2a
C1sin(μ ln|x|) + C2cos(μ ln|x|)
if (1– a)2<4b, where μ = 12|(1– a)2–4b|1 2.
13. x2y
xx + xy x + (x2– ν2)y = 0.
Bessel equation.
1◦ Let ν be an arbitrary noninteger Then the general solution is given by
y = C1J ν (x) + C2Y ν (x), (1)
where J ν (x) and Y ν (x) are the Bessel functions of the first and second kind:
J ν (x) =
∞
k=0
(–1)k (x/2)ν+2k
k! Γ(ν + k +1), Y ν (x) =
J ν (x) cos πν – J– ν (x)
Solution (1) is denoted by y = Z ν (x), which is referred to as the cylindrical function The functions J ν (x) and Y ν (x) can be expressed in terms of definite integrals (with
x>0):
πJ ν (x) = π
0 cos(x sin θ – νθ) dθ – sin πν
∞
0 exp(–x sinh t – νt) dt,
πY ν (x) = π
0 sin(x sin θ – νθ) dθ –
∞
0 (e νt + e
–νt cos πν)e–x sinh t dt.
2◦ In the case ν = n +1
2, where n =0, 1, 2, , the Bessel functions are expressed in terms
of elementary functions:
J n+1(x) = 2
π x
n+1
–1
x
d dx
n sin x
x , J–n–1(x) = 2
π x
n+11
x
d dx
n cos x
x ,
Y n+12(x) = (–1)n+1J–n–12(x).
The Bessel functions are described in Section 18.6 in detail
14. x2y
xx + xy x – (x2+ ν2)y = 0.
Modified Bessel equation It can be reduced to equation T5.2.1.13 by means of the
substi-tution x = i ¯x (i2 = –1)
Solution:
y = C1I ν (x) + C2K ν (x),
where I ν (x) and K ν (x) are modified Bessel functions of the first and second kind:
I ν (x) =
∞
k=0
(x/2)2k+ν
π
2
I–ν (x) – I ν (x)
sin πν .
The modified Bessel functions are described in Subsection 18.7 in detail
Trang 415. x2y
xx + axy x + (bx n + c)y = 0, n≠0.
The case b =0corresponds to the Euler equation T5.2.1.12
For b≠ 0, the solution is
y = x1–2a*
C1J ν
n
√
b x n
2
+ C2Y ν
n
√
b x n
2+
,
where ν = n1
(1– a)2–4c; J ν (z) and Y ν (z) are the Bessel functions of the first and
second kind
16. x2y
xx + axy x + x n (bx n + c)y = 0.
The substitution ξ = x n leads to an equation of the form T5.2.1.11:
n2ξy
ξξ + n(n –1+ a)y ξ + (bξ + c)y =0
17. x2y
xx + (ax + b)y x + cy = 0.
The transformation x = z–1, y = z k e z w, where k is a root of the quadratic equation
k2+ (1– a)k + c =0, leads to an equation of the form T5.2.1.11:
zw
zz+ [(2– b)z +2k+2– a]w z+ [(1– b)z +2k+2– a – bk]w =0
18. (1 – x2)y
xx – 2xy x + n(n + 1)y = 0, n = 0, 1, 2,
Legendre equation.
The solution is given by
y = C1P n (x) + C2Q n (x),
where the Legendre polynomials P n (x) and the Legendre functions of the second kind
Q n (x) are given by the formulas
P n (x) = 1
n!2n
d n
dx n (x2–1)n, Q n (x) = 1
2P n (x) ln
1+ x
1– x –
n
m=1
1
m P m–1(x)P n–m (x).
The functions P n = P n (x) can be conveniently calculated using the recurrence relations
P0(x) =1, P1(x) = x, P2(x) =1
2(3x2–1), , P n+1(x) =
2n+1
n+1 xP n (x)–
n n+1P n–1(x).
Three leading functions Q n = Q n (x) are
Q0(x) = 1
2ln
1+ x
1– x, Q1(x) =
x
2 ln
1+ x
1– x –1, Q2(x) = 3x2–1
4 ln
1+ x
1– x –
3
2x.
The Legendre polynomials and the Legendre functions are described in Subsection 18.11.1 in more detail
Trang 519. (1 – x2)y
xx – 2xy x + ν(ν + 1)y = 0.
Legendre equation; ν is an arbitrary number The case ν = n where n is a nonnegative
integer is considered in T5.2.1.18
The substitution z = x2 leads to the hypergeometric equation Therefore, with|x|<1
the solution can be written as
y = C1F
–ν
2,
1+ ν
2 ,
1
2; x
+ C2xF
2 , 1+
ν
2,
3
2; x
,
where F (α, β, γ; x) is the hypergeometric series (see T5.2.1.22).
The Legendre equation is discussed in Subsection 18.11.3 in more detail
20. (ax2+ b)y xx + axy x + cy = 0.
The substitution z = √ dx
ax2+ b leads to a constant coefficient linear equation: y zz +cy =0
21. (1 – x2)y xx + (ax + b)y x + cy = 0.
1◦ The substitution 2z=1+ x leads to the hypergeometric equation T5.2.1.22:
z(1– z)y zz + [az + 12(b – a)]y z + cy =0
2◦ For a = –2m–3, b =0, and c = λ, the Gegenbauer functions are solutions of the equation.
22. x(x – 1)y xx + [(α + β + 1)x – γ]y x + αβy = 0.
Gaussian hypergeometric equation For γ≠ 0, –1, –2, –3, , a solution can be expressed
in terms of the hypergeometric series:
F (α, β, γ; x) =1+
∞
k=1
(α) k (β) k (γ) k
x k
k!, (α) k = α(α +1) (α + k –1),
which, a fortiori, is convergent for|x|<1
For γ > β >0, this solution can be expressed in terms of a definite integral:
F (α, β, γ; x) = Γ(γ)
Γ(β) Γ(γ – β)
1
0 t β–
1(1– t) γ–β–1(1– tx)–α dt,
whereΓ(β) is the gamma function.
If γ is not an integer, the general solution of the hypergeometric equation has the form:
y = C1F(α, β, γ; x) + C2x1 –γ F (α – γ +1, β – γ +1, 2– γ; x).
In the degenerate cases γ =0, –1, –2, –3, , a particular solution of the hypergeometric equation corresponds to C1 =0 and C2 = 1 If γ is a positive integer, another particular solution corresponds to C1=1and C2=0 In both these cases, the general solution can be constructed by means of formula (2) given in the preliminary remarks at the beginning of Section T5.2
Table T5.2 gives the general solutions of the hypergeometric equation for some values
of the determining parameters
The hypergeometric functions F (α, β, γ; x) are discussed in Section 18.10 in detail.
Trang 6TABLE T5.2 General solutions of the hypergeometric equation for some values of the determining parameters
|x| –γ|x– 1|γ–β–1dx
α α+12 2α+ 1 C1 1 +√
1– x – 2α + C2x–2α 1 +√
1– x 2α
α α–12 12 C1 1 +√
x 1 – 2α + C2 1 –√
x 1 – 2α
x
*
C1 1 +√
x 1– 2α + C2 1 –√
x 1– 2α+
C1+ C2
|x|γ–2 |x– 1|β–γ dx
C1+ C2
|x| –α |x– 1|β–1dx
C1+ C2
|x|α–1 |x– 1| –βdx
23. (1 – x2 ) 2y
xx – 2x(1 – x2)y
x + [ν(ν + 1)(1 – x2) – μ2]y = 0.
Legendre equation, ν and μ are arbitrary parameters.
The transformation x =1–2ξ, y =|x2–1|μ/2w leads to the hypergeometric equation
T5.2.1.22:
ξ(ξ –1)w ξξ + (μ +1)(1–2ξ)w ξ + (ν – μ)(ν + μ +1)w =0
with parameters α = μ – ν, β = μ + ν +1, γ = μ +1
In particular, the original equation is integrable by quadrature if ν = μ or ν = –μ –1
24. (x – a)2(x – b)2y
xx – cy = 0, a ≠b.
The transformation ξ = lnx – a
x – b
, y = (x–b)η leads to a constant coefficient linear equation:
(a – b)2(η ξξ – η ξ ) – cη =0 Therefore, the solution is as follows:
y = C1|x – a|( 1 +λ)/2|x – b|( 1 –λ)/2+ C
2|x – a|( 1 –λ)/2|x – b|( 1 +λ)/2,
where λ2=4c(a – b)– 2+1 ≠ 0
25. (ax2+ bx + c)2y
xx + Ay = 0.
The transformation ξ = dx
ax2+ bx + c , w =
y
|ax2+ bx + c| leads to a constant coefficient
linear equation of the form T5.2.1.1: w ξξ + (A + ac – 14b2)w =0
26. x2(ax n – 1)y
xx + x(apx n + q)y x + (arx n + s)y = 0.
Find the roots A1, A2and B1, B2of the quadratic equations
A2– (q +1)A – s =0, B2– (p –1)B + r =0
Trang 7and define parameters c, α, β, and γ by the relations
c = A1, α = (A1+ B1)n–1, β = (A1+ B2)n–1, γ =1+ (A1– A2)n–1
Then the solution of the original equation has the form y = x c u(ax n ), where u = u(z) is the general solution of the hypergeometric equation T5.2.1.22: z(z–1)u zz +[(α+β+1)z–γ]u z+
αβu=0
T5.2.2 Equations Involving Exponential and Other Functions
1. y
xx + ae λx y = 0, λ≠0.
Solution: y = C1J0(z) + C2Y0(z), where z =2λ–1√
a e λx/2; J0(z) and Y0(z) are Bessel
functions
2. y
xx + (ae x – b)y = 0.
Solution: y = C1J2√ b 2√ a e x/2
+ C2Y2√ b 2√ a e x/2
, where J ν (z) and Y ν (z) are Bessel
functions
3. y
xx – (ae2λx + be λx + c)y = 0.
The transformation z = e λx , w = z–k y, where k = √
c/λ, leads to an equation of the form T5.2.1.11: λ2zw
zz + λ2(2k+1)w z – (az + b)w =0
4. y
xx + ay x + be2ax y= 0.
The transformation ξ = e ax , u = ye ax leads to a constant coefficient linear equation of the
form T5.2.1.1: u ξξ + ba–2u=0
5. y
xx – ay x + be2ax y= 0.
The substitution ξ = e ax leads to a constant coefficient linear equation of the form T5.2.1.1:
y
ξξ + ba–2y=0
6. y
xx + ay x + (be λx + c)y = 0.
Solution: y = e–ax/2
C1J ν 2λ–1√
b e λx/2
+C2Y ν 2λ–1√
b e λx/2
, where ν = λ1√
a2–4c;
J ν (z) and Y ν (z) are Bessel functions.
7. y
xx – (a – 2q cosh 2x)y = 0.
Modified Mathieu equation The substitution x = iξ leads to the Mathieu equation T5.2.2.8:
y
ξξ + (a –2qcos2ξ)y =0
For eigenvalues a = a n (q) and a = b n (q), the corresponding solutions of the modified Mathieu
equation are
Ce2n+p (x, q) = ce2n+p (ix, q) =
∞
k=0
A2n+p
2k+pcosh[(2k + p)x],
Se2n+p (x, q) = –i se2n+p (ix, q) =
∞
k=0
B2n+p
2k+psinh[(2k + p)x], where p can be either 0 or 1, and the coefficients A22n+p k+p and B22k+p n+pare specified in T5.2.2.8 The functions Ce2n+p (x, q) and Se2n+p (x, q) are discussed in Section 18.16 in more
detail
... T5.2.1.13) The resultsof solving the original equation are presented in Table T5.1
TABLE T5.1 Solutions of equation T5.2.1.11 for different values of the determining parameters... (α + k –1),
which, a fortiori, is convergent for< /i>|x|<1
For γ > β >0, this solution can be expressed in terms of a definite integral:
F... –2, –3, , a particular solution of the hypergeometric equation corresponds to C1 =0 and C2 = 1 If γ is a positive integer, another particular solution