Page 347 - Handbook Of Integral Equations

For equations 4.9.41–4.9.46, only particular solutions are given. To obtain the general solution,
one must add the particular solution to the general solution of the corresponding homogeneous
equation 4.9.40.

b
41. y(x)+ f(t)y(x – t) dt = Ax + B.
a
A solution:
y(x)= px + q,
where the coefficients p and q are given by
b b
A AI 1 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
42. y(x)+ f(t)y(x – t) dt = Ae λx .
a
A solution:
A λx b
y(x)= e , B =1 + f(t) exp(–λt) dt.
B a
The general solution of the integral equation is the sum of the specified particular solution
and the general solution of the homogeneous equation 4.9.40.

b

43. y(x)+ f(t)y(x – t) dt = A sin(λx).
a
A solution:
AI c AI s
y(x)= sin(λx)+ cos(λx),
I + I 2 I + I 2
2
2
c s c s
where the coefficients I c and I s are given by

b b
I c =1 + f(t) cos(λt) dt, I s = f(t) sin(λt) dt.
a a
b

44. y(x)+ f(t)y(x – t) dt = A cos(λx).
a
A solution:
AI s AI c
y(x)= – sin(λx)+ cos(λx),
I + I 2 I + I 2
2
2
c s c s
where the coefficients I c and I s are given by
b b

I c =1 + f(t) cos(λt) dt, I s = f(t) sin(λt) dt.
a a

b
45. y(x)+ f(t)y(x – t) dt = e µx (A sin λx + B cos λx).
a
A solution:
µx
y(x)= e (p sin λx + q cos λx),
where the coefficients p and q are given by
AI c – BI s AI s + BI c
p = , q = ,
2
2
I + I 2 I + I 2
c s c s

b b
I c =1 + f(t)e –µt cos(λt) dt, I s = f(t)e –µt sin(λt) dt.
a a
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 326