Page 389 - Handbook Of Integral Equations

x

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

y (x)+ f x, y(x) + λ sinh[λ(x – t)]f t, y(t) dt = g (x), (1)
x
x
a
x
2
y (x)+ f x, y(x) x + λ cosh[λ(x – t)]f t, y(t) dt = g (x). (2)

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

y xx + f(x, y) x – λ y + λ g(x) – g (x)=0. (3)
xx
By setting x = a in the original equation and in (1), we obtain the initial conditions for y = y(x):

y(a)= g(a), y (a)= g (a) – f a, g(a) . (4)
x
x
Equation (3) under conditions (4) defines the solution of the original integral equation.
x

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

y (x)+ λ cosh[λ(x – t)]f t, y(t) dt = g (x), (1)

x x
a
x

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

xx
xx
a
Eliminating the integral term from (2) with the aid of the original equation, we arrive at
the second-order nonlinear ordinary differential equation
2
2
y + λf(x, y) – λ y + λ g(x) – g (x) = 0. (3)

xx xx
By setting x = a in the original equation and in (1), we obtain the initial conditions for y = y(x):
y(a)= g(a), y (a)= g (a). (4)

x x
Equation (3) under conditions (4) defines the solution of the original integral equation. For
the exact solutions of the second-order differential equation (3) with various f(x, y) and g(x),
see A. D. Polyanin and V. F. Zaitsev (1995), and V. F. Zaitsev and A. D. Polyanin (1994).
x

24. y(x)+ cos[λ(x – t)]f t, y(t) dt = g(x).
a
Differentiating the equation with respect to x twice yields
x

y (x)+ f x, y(x) – λ sin[λ(x – t)]f t, y(t) dt = g (x), (1)

x
x
a
x
2

y (x)+ f x, y(x) x – λ cos[λ(x – t)]f t, y(t) dt = g (x). (2)
xx
xx
a
Eliminating the integral term from (2) with the aid of the original equation, we arrive at
the second-order nonlinear ordinary differential equation
2 2
y xx + f(x, y) x + λ y – λ g(x) – g (x)=0. (3)

xx
By setting x = a in the original equation and in (1), we obtain the initial conditions for y = y(x):

y(a)= g(a), y (a)= g (a) – f a, g(a) . (4)
x
x
Equation (3) under conditions (4) defines the solution of the original integral equation.
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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