Page 349 - Handbook Of Integral Equations

n
8 .For g(x)= e µx A k cos(λ k x), a solution of the equation has the form
◦
k=1
n n

y(x)= e µx B k cos(λ k x)+ e µx 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.
n
9 .For g(x)= e µx A k sin(λ k x), a solution of the equation has the form
◦
k=1
n n

y(x)= e µx B k cos(λ k x)+ e µx 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.
n
◦
10 .For g(x) = cos(λx) A k exp(µ k x), a solution of the equation has 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.

n
11 .For g(x) = sin(λx) A k exp(µ k x), a solution of the equation has 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
47. y(x)+ f(t)y(x + βt) dt = Ax + B.
a
A solution:*
y(x)= px + q,
where

A B AI 1 β b b
p = , q = – 2 , I 0 = f(t) dt, I 1 = tf(t) dt.
1+ I 0 1+ I 0 (1 + I 0 ) a a
b
48. y(x)+ f(t)y(x + βt) dt = Ae λx .
a
A solution:
A λx b
y(x)= e , B =1 + f(t) exp(λβt) dt.
B a
* In the equations below that contain y(x + βt), β > 0, in the integrand, the arguments can have, for example, the domain
(a) 0 ≤ x < ∞,0 ≤ t < ∞ for a = 0 and b = ∞ or (b) a ≤ t ≤ b,0 ≤ x < ∞ for a and b such that 0 ≤ a < b < ∞. Case (b) is
a special case of (a) if f(t) is nonzero only on the interval a ≤ t ≤ b.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 328