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are satisfied. Here the constants B k(i) form p linearly independent solutions of the homogeneous
system of algebraic equations which is the transpose of system (15). In this case, every solution of
Eq. (7) has the form
p

y(x)= y 0 (x)+ C i y i (x), (17)
i=1
where y 0 (x) is a particular solution of the nonhomogeneous equation (7) and the sum represents the
◦
general solution of the corresponding homogeneous equation (see item 2 ). In particular, if f(x) ≠ 0
but all f k are zero, we have
p

y(x)= f(x)+ C i y i (x). (18)
i=1
Remark. When studying Fredholm equations of the second kind with degenerate kernel, it is
useful for the reader to be acquainted with equations 4.9.18 and 4.9.20 of the first part of the book.
Example. Let us solve the integral equation
π

2
y(x) – λ (x cos t + t sin x + cos x sin t)y(t) dt = x, –π ≤ x ≤ π. (19)
–π
Let us denote
π π π
2
A 1 = y(t) cos tdt, A 2 = t y(t) dt, A 3 = y(t) sin tdt, (20)
–π –π –π
where A 1 , A 2 , and A 3 are unknown constants. Then Eq. (19) can be rewritten in the form
y(x)= A 1 λx + A 2 λ sin x + A 3 λ cos x + x. (21)
On substituting the expression (21) into relations (20), we obtain
π

A 1 = (A 1 λt + A 2 λ sin t + A 3 λ cos t + t) cos tdt,
–π
π

2
A 2 = (A 1 λt + A 2 λ sin t + A 3 λ cos t + t)t dt,
–π
π

A 3 = (A 1 λt + A 2 λ sin t + A 3 λ cos t + t) sin tdt.
–π
On calculating the integrals occurring in these equations, we obtain the following system of algebraic equations for the
unknowns A 1 , A 2 , and A 3 :
A 1 – λπA 3 =0,
A 2 +4λπA 3 =0, (22)
–2λπA 1 – λπA 2 + A 3 =2π.
The determinant of this system is

1 0
–λπ
0 1 4λπ = 1+2λ π ≠ 0.
2 2
∆(λ)=
–2λπ –λπ 1

Thus, system (22) has the unique solution
2λπ 2 8λπ 2 2π
A 1 = , A 2 = – , A 3 = .
2 2
2 2
2 2
1+2λ π 1+2λ π 1+2λ π
On substituting the above values of A 1 , A 2 , and A 3 into (21), we obtain the solution of the original integral equation:
2λπ
y(x)= (λπx – 4λπ sin x + cos x)+ x.
2 2
1+2λ π
•
References for Section 11.2: S. G. Mikhlin (1960), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971),
I. S. Gradshteyn and I. M. Ryzhik (1980), A. J. Jerry (1985), A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1986,
1988).
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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