Page 426 - Handbook Of Integral Equations

b
λx
18. y(x – t)f t, y(t) dt = Ae .
a
λx
This equation has solutions of the form y(x)= pe , where p is some constant.
b

19. y(x – t)f t, y(t) dt = A cos λx.
a
This equation has solutions of the form y(x)= p sin λx + q cos λx, where p and q are some
constants.
b
–µx
20. y(x – t)f t, y(t) dt = e (A cos λx + B sin λx).
a
This equation has solutions of the form y(x)= e –µx (p sin λx + q cos λx), where p and q are
some constants.

b

6.8-2. Equations of the Form y(x)+ K(x, t)G y(t) dt = F (x)
a
b
2
21. y(x)+ |x – t|f y(t) dt = Ax + Bx + C.
a
2
This is a special case of equation 6.8.35 with f(t, y)= f(y) and g(x)= Ax + Bx + C.
The function y = y(x) obeys the second-order autonomous differential equation
y xx +2f(y)=2A,
whose solution can be represented in an implicit form:

y u
du

= ±(x – a), F(u, v)= f(t) dt, (1)

2
w +4A(u – y a ) – 4F(u, y a ) v
y 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
2
2
y a + y b – (a – b)w a =(b +2ab – a )A +2bB +2C,
w a + w b =2(a + b)A +2B,
2
2
w = w +4A(y b – y a ) – 4F(y b , y a ), (2)
b a

du
y b
= ±(b – a).

2
w +4A(u – y a ) – 4F(u, y a )
y a a
Here the first equation is obtained from the second condition of (5) in 6.8.35, the second
equation is obtained from condition (6) in 6.8.35, and the third and fourth equations are
consequences of (1).
Each solution of system (2) generates a solution of the integral equation.
b

22. y(x)+ e λ|x–t| f y(t) dt = A + Be λx + Ce –λx .
a
This is a special case of equation 6.8.36 with f(t, y)= f(y) and g(x)= A + Be λx + Ce –λx .
The function y = y(x) satisfies the second-order autonomous differential equation
2
2

y xx +2λf(y) – λ y = –λ A, (1)
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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