Page 175 - Handbook Of Integral Equations

x

28. y(x)+ A sin(kt)+ B + AB(x – t) sin(kt) y(t) dt = f(x).
a
This is a special case of equation 2.9.8 with λ = B and g(t)= A sin(kt).
Solution:
x
y(x)= f(x)+ R(x, t)f(t) dt,
a
G(t) B 2 x B(t–s) A
R(x, t)= –[A sin(kt)+ B] + e G(s) ds, G(x)=exp – cos(kx) .
G(x) G(x) t k

∞
√
29. y(x)+ A sin λ t – x y(t) dt = f(x).
x
√
This is a special case of equation 2.9.62 with K(x)= A sin λ –x .
2.5-3. Kernels Containing Tangent

x
30. y(x) – A tan(λx)y(t) dt = f(x).
a
This is a special case of equation 2.9.2 with g(x)= A tan(λx) and h(t)=1.
Solution:
x

A/λ

y(x)= f(x)+ A tan(λx) cos(λt) f(t) dt.
a cos(λx)
x
31. y(x) – A tan(λt)y(t) dt = f(x).
a
This is a special case of equation 2.9.2 with g(x)= A and h(t) = tan(λt).
Solution:
x A/λ
y(x)= f(x)+ A tanh(λt) cos(λt) f(t) dt.
a cos(λx)
x

32. y(x)+ A tan(λx) – tan(λt) y(t) dt = f(x).
a
This is a special case of equation 2.9.5 with g(x)= A tan(λx).
Solution: x
1

y(x)= f(x)+ Y (x)Y (t) – Y (x)Y (t) f(t) dt,

2
2
1
1
W a
where Y 1 (x), Y 2 (x) is a fundamental system of solutions of the second-order linear ordinary
2
differential equation cos (λx)Y + AλY =0, W is the Wronskian, and the primes stand for
xx
the differentiation with respect to the argument specified in the parentheses.
As shown in A. D. Polyanin and V. F. Zaitsev (1995, 1996), the functions Y 1 (x) and Y 2 (x)
can be expressed via the hypergeometric function.
x tan(λx)
33. y(x) – A y(t) dt = f(x).
a tan(λt)
Solution:
x tan(λx)
y(x)= f(x)+ A e A(x–t) f(t) dt.
a tan(λt)
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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