Page 440 - Handbook Of Integral Equations

9 .For g(x)= e µx n A k sin(λ k x), the equation has a solution of the form
◦
k=1
n n
µx
µx
y(x)= e B k cos(λ k x)+ e C k sin(λ k x),
k=1 k=1
where the constants B k and C k can be found by the method of undetermined coefficients.

10 .For g(x) = cos(λx) n A k exp(µ k x), the equation has a solution of the form
◦
k=1
n n

y(x) = cos(λx) B k exp(µ k x) + sin(λx) B k exp(µ k x),
k=1 k=1

where the constants B k and C k can be found by the method of undetermined coefficients.

◦
11 .For g(x) = sin(λx) n A k exp(µ k x), the equation has a solution of the form
k=1
n n

y(x) = cos(λx) B k exp(µ k x) + sin(λx) B k exp(µ k x),
k=1 k=1
where the constants B k and C k can be found by the method of undetermined coefficients.

b

63. y(x)+ y(x + βt)f t, y(t) dt = Ax + B.
a
A solution:
y(x)= px + q, (1)

where p and q are roots of the following system of algebraic (or transcendental) equations:

b

p + p f(t, pt + q) dt – A =0,
a
b (2)

q + (βpt + q)f(t, pt + q) dt – B =0.
a
Different solutions of system (2) generate different solutions (1) of the integral equation.
b

λx
64. y(x)+ y(x + βt)f t, y(t) dt = Ae .
a
Solutions:
λx
y(x)= k n e ,
where k n are roots of the algebraic (or transcendental) equation

b
λt βλt
k + kF(k) – A =0, F(k)= f t, ke e dt.
a

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 420