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x

30. y(x)+ sinh[λ(x – t)]g(t)y(t) dt = f(x).
a
1 . Differentiating the equation with respect to x twice yields
◦
x
y (x)+ λ cosh[λ(x – t)]g(t)y(t) dt = f (x), (1)

x
x
a
x

y (x)+ λg(x)y(x)+ λ 2 sinh[λ(x – t)]g(t)y(t) dt = f (x). (2)

xx xx
a
Eliminating the integral term from (2) with the aid of the original equation, we arrive at
the second-order linear ordinary differential equation
2

y + λ g(x) – λ y = f (x) – λ f(x). (3)
xx
xx
By setting x = a in the original equation and (1), we obtain the initial conditions for y = y(x):

y(a)= f(a), y (a)= f (a). (4)

x
x
For exact solutions of second-order linear ordinary differential equations (3) with vari-
ous g(x), see E. Kamke (1977), G. M. Murphy (1960), and A. D. Polyanin and V. F. Zaitsev
(1995, 1996).
2 . Let y 1 = y 1 (x) and y 2 = y 2 (x) be two linearly independent solutions (y 1 /y 2 /≡ const) of
◦

the homogeneous differential equation y xx + λ g(x) – λ y = 0, which follows from (3) for

f(x) ≡ 0. In this case, the Wronskian is a constant:
W = y 1 (y 2 ) – y 2 (y 1 ) ≡ const .

x
x
The solution of the nonhomogeneous equation (3) under conditions (4) with arbitrary f = f(x)
has the form x
λ
y(x)= f(x)+ y 1 (x)y 2 (t) – y 2 (x)y 1 (t) g(t)f(t) dt (5)
W a
and determines the solution of the original integral equation.
◦
3 . Given only one nontrivial solution y 1 = y 1 (x) of the linear homogeneous differential

equation y +λ g(x)–λ y =0, one can obtain the solution of the nonhomogeneous equation (3)
xx
under the initial conditions (4) by formula (5) with
x
dξ
W =1, y 2 (x)= y 1 (x) 2 ,
b y (ξ)
1
where b is an arbitrary number.
x

31. y(x)+ sinh[λ(x – t)]g(x)h(t)y(t) dt = f(x).
a
The substitution y(x)= g(x)u(x) leads to an equation of the form 2.9.30:
x
u(x)+ sinh[λ(x – t)]g(t)h(t)u(t) dt = f(x)/g(x).
a

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 180