Page 132 - Handbook Of Integral Equations

x

2 2
17. y(x)+ Ax – At + Bx – Ct + D y(t) dt = f(x).
a
2
2
This is a special case of equation 2.9.6 with g(x)= Ax + Bx + D and h(t)= –At – Ct.
Solution:
x
∂ 2 Y 1 (x)Y 2 (t) – Y 2 (x)Y 1 (t)
y(x)= f(x)+ f(t) dt.
∂x∂t W(t)
a
Here Y 1 (x), Y 2 (x) is a fundamental system of solutions of the second-order homogeneous

ordinary differential equation Y + (B – C)x + D Y +(2Ax + B)Y = 0 (see A. D. Polyanin

xx x
and V. F. Zaitsev (1996) for details about this equation):
1 1 2 1 1 2
Y 1 (x)=exp(–kx)Φ α, ; (C – B)z , Y 2 (x) = exp(–kx)Ψ α, ; (C – B)z ,
2 2 2 2
√
2π(C – B) 2 2A
W(x)= – exp 1 (C – B)z – 2kx , k = ,
Γ(α) 2 B – C
2
4A +2AD(C – B)+ B(C – B) 2 4A +(C – B)D
α = – , z = x – ,
2(C – B) 3 (C – B) 2

where Φ α, β; x and Ψ α, β; x are degenerate hypergeometric functions and Γ(α)isthe
gamma function.
x

18. y(x) – Ax + B +(Cx + D)(x – t) y(t) dt = f(x).
a
This is a special case of equation 2.9.11 with g(x)= Ax + B and h(x)= Cx + D.
Solution with A ≠ 0:
x
f(t)

y(x)= f(x)+ Y (x)Y 1 (t) – Y (x)Y 2 (t) dt.
2 1
a W(t)
Here Y 1 (x), Y 2 (x) is a fundamental system of solutions of the second-order homogeneous
ordinary differential equation Y xx – (Ax + B)Y – (Cx + D)Y = 0 (see A. D. Polyanin and

x
V. F. Zaitsev (1996) for details about this equation):
1 1 2 1 1 2
Y 1 (x) = exp(–kx)Φ α, ; Az , Y 2 (x) = exp(–kx)Ψ α, ; Az ,
2 2 2 2
√
–1 1 2
W(x)= – 2πA Γ(α) exp Az – 2kx , k = C/A,
2
–2
1
2
2
–3
α = (A D – ABC – C )A , z = x +(AB +2C)A ,
2

where Φ α, β; x and Ψ α, β; x are degenerate hypergeometric functions, Γ(α) is the gamma
function.
x

19. y(x)+ At + B +(Ct + D)(t – x) y(t) dt = f(x).
a
This is a special case of equation 2.9.12 with g(t)= –At – B and h(t)= –Ct – D.
Solution with A ≠ 0:
x f(t)

y(x)= f(x) – Y 1 (x)Y (t) – Y (t)Y 2 (x) dt.

2
1
a W(x)
Here Y 1 (x), Y 2 (x) is a fundamental system of solutions of the second-order homogeneous

ordinary differential equation Y – (Ax + B)Y – (Cx + D)Y = 0 (see A. D. Polyanin and
xx x
V. F. Zaitsev (1996) for details about this equation):
1 1 2 1 1 2
Y 1 (x) = exp(–kx)Φ α, ; Az , Y 2 (x) = exp(–kx)Ψ α, ; Az ,
2 2 2 2
√
–1 1 2
W(x)= – 2πA Γ(α) exp Az – 2kx , k = C/A,
2
–2
2
2
1
–3
α = (A D – ABC – C )A , z = x +(AB +2C)A ,
2

where Φ α, β; x and Ψ α, β; x are degenerate hypergeometric functions and Γ(α)isthe
gamma function.
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 111