Page 280 - Handbook Of Integral Equations

b

m
36. y(x) – λ (Ax m+1 + Bx t + Cx m + D)y(t) dt = f(x), m =1, 2, ...
a
m
m
This is a special case of equation 4.9.18 with g 1 (x)= Ax m+1 +Cx +D, h 1 (t)=1, g 2 (x)= x ,
and h 2 (t)= Bt.
Solution:
m
m
y(x)= f(x)+ λ[A 1 (Ax m+1 + Cx + D)+ A 2 x ],
where A 1 and A 2 are the constants determined by the formulas presented in 4.9.18.
b
37. y(x) – λ (Axt m + Bt m+1 + Ct m + D)y(t) dt = f(x), m =1, 2, ...
a
m
This is a special case of equation 4.9.18 with g 1 (x)= x, h 1 (t)= At , g 2 (x) = 1, and
m
h 2 (t)= Bt m+1 + Ct + D.
Solution:
y(x)= f(x)+ λ(A 1 x + A 2 ),
where A 1 and A 2 are the constants determined by the formulas presented in 4.9.18.

b
n n
m m
38. y(x) – λ (Ax t + Bx t )y(t) dt = f(x), n, m =1, 2, ... , n ≠ m.
a
m
n
This is a special case of equation 4.9.14 with g(x)= x and h(t)= t .
Solution:
n
m
y(x)= f(x)+ λ(A 1 x + A 2 x ),
where A 1 and A 2 are the constants determined by the formulas presented in 4.9.14.
b
m n
n m
39. y(x) – λ (Ax t + Bx t )y(t) dt = f(x), n, m =1, 2, ... , n ≠ m.
a
n
m
This is a special case of equation 4.9.17 with g(x)= x and h(t)= t .
Solution:
m
n
y(x)= f(x)+ λ(A 1 x + A 2 x ),
where A 1 and A 2 are the constants determined by the formulas presented in 4.9.17.
b
m
40. y(x) – λ (x – t) y(t) dt = f(x), m =1, 2, ...
a
This is a special case of equation 4.9.19 with g(x)= x and h(t)= –t.
b

m
41. y(x) – λ (Ax + Bt) y(t) dt = f(x), m =1, 2, ...
a
This is a special case of equation 4.9.19 with g(x)= Ax and h(t)= Bt.

b
k
42. y(x)+ A |x – t|t y(t) dt = f(x).
a
k
This is a special case of equation 4.9.36 with g(t)= At . Solving the integral equation
k
is reduced to solving the ordinary differential equation y xx +2Ax y = f (x), the general

xx
solution of which can be expressed via Bessel functions or modified Bessel functions (the
boundary conditions are given in 4.9.36).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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