Page 180 - Handbook Of Integral Equations

x

21. y(x) – A arccot(λt)y(t) dt = f(x).
a
This is a special case of equation 2.9.2 with g(x)= A and h(t) = arccot(λt).

x
arccot(λx)
22. y(x) – A y(t) dt = f(x).
a arccot(λt)
Solution:
x
arccot(λx)
y(x)= f(x)+ A e A(x–t) f(t) dt.
arccot(λt)
a
x arccot(λt)
23. y(x) – A y(t) dt = f(x).
a arccot(λx)
Solution:
x
arccot(λt)
y(x)= f(x)+ A e A(x–t) f(t) dt.
a arccot(λx)
∞

24. y(x)+ A arccot[λ(t – x)]y(t) dt = f(x).
x
This is a special case of equation 2.9.62 with K(x)= A arccot(–λx).

x

25. y(x) – A arccot(kx)+ B – AB(x – t) arccot(kx) y(t) dt = f(x).
a
This is a special case of equation 2.9.7 with λ = B and g(x)= A arccot(kx).

x

26. y(x)+ A arccot(kt)+ B + AB(x – t) arccot(kt) y(t) dt = f(x).
a
This is a special case of equation 2.9.8 with λ = B and g(t)= A arccot(kt).

2.7. Equations Whose Kernels Contain Combinations of
Elementary Functions

2.7-1. Kernels Containing Exponential and Hyperbolic Functions

x
1. y(x)+ A e µ(x–t) cosh[λ(x – t)]y(t) dt = f(x).
a
Solution:
x

y(x)= f(x)+ R(x – t)f(t) dt,
a

A 2
1 2 1 2
R(x)=exp (µ – A)x sinh(kx) – A cosh(kx) , k = λ + A .
2 2k 4

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 159