Page 427 - Handbook Of Integral Equations

whose solution can be written in an implicit form:
y u
du

= ±(x–a), F(u, v)= f(t) dt, (2)

2 2 2 2 2
w +λ (u –y )–2Aλ (u–y a )–4λF(u, y a ) v
y a a a

where y a = y(a) and w a = y (a) are constants of integration. These constants, as well as
x
the unknowns y b = y(b) and w b = y (b), are determined by the algebraic (or transcendental)

x
system
λa
w a + λy a = Aλ +2Bλe ,
w b – λy b = –Aλ – 2Cλe –λb ,
2
2
2
2
2
2
w = w + λ (y – y ) – 2Aλ (y b – y a ) – 4λF(y b , y a ), (3)
b a b a

y b
du
= ±(b – a).

2 2 2 2 2
w + λ (u – y ) – 2Aλ (u – y a ) – 4λF(u, y a )
y a a a
Here the first and second equations are obtained from conditions (5) in 6.8.86, and the third
and fourth equations are consequences of (2).
Each solution of system (3) generates a solution of the integral equation.
b

23. y(x)+ e λ|x–t| f y(t) dt = β cosh(λx).
a
1
This is a special case of equation 6.8.22 with A = 0 and B = C = β.
2
b

24. y(x)+ e λ|x–t| f y(t) dt = β sinh(λx).
a
1
1
This is a special case of equation 6.8.22 with A =0, B = β, and C = – β.
2 2
b

25. y(x)+ sinh λ|x – t| f y(t) dt = A + B cosh(λx)+ C sinh(λx).
a
This is a special case of equation 6.8.37 with f(t, y)= f(y) and g(x)= A + B cosh(λx)+
C sinh(λx).
The function y = y(x) satisfies the second-order autonomous differential equation
2
2
y +2λf(y) – λ y = –λ A,
xx
whose solution can be represented in an implicit form:
y u
du
= ±(x – a), F(u, v)= f(t) dt,

2
2
w + λ (u – y ) – 2Aλ (u – y a ) – 4λF(u, y a ) v
2
2
2
y a a a
where y a = y(a) and w a = y (a) are constants of integration, which can be determined from

x
the boundary conditions (5) in 6.8.37.
b

26. y(x)+ sin λ|x – t| f y(t) dt = A + B cos(λx)+ C sin(λx).
a
This is a special case of equation 6.8.38 with f(t, y)=f(y) and g(x)=A+B cos(λx)+C sin(λx).
The function y = y(x) satisfies the second-order autonomous differential equation
2
2
y +2λf(y)+ λ y = λ A,
xx
whose solution can be represented in an implicit form:
y u
du
= ±(x – a), F(u, v)= f(t) dt,

2 2 2 2 2
w – λ (u – y )+2Aλ (u – y a ) – 4λF(u, y a ) v
y a a a

where y a = y(a) and w a = y (a) are constants of integration, which can be determined from
x
the boundary conditions (5) in 6.8.38.
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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