Page 435 - Handbook Of Integral Equations

b

50. y(x)+ y(xt)f t, y(t) dt = A cos(ln x).
a
A solution:
y(x)= p cos(ln x)+ q sin(ln x),
where p and q are roots of the following system of algebraic (or transcendental) equations:

b

p + p cos(ln t)+ q sin(ln t) f t, p cos(ln t)+ q sin(ln t) dt = A,
a
b

q + q cos(ln t) – p sin(ln t) f t, p cos(ln t)+ q sin(ln t) dt =0.
a
b

51. y(x)+ y(xt)f t, y(t) dt = A sin(ln x).
a
A solution:
y(x)= p cos(ln x)+ q sin(ln x),
where p and q are roots of the following system of algebraic (or transcendental) equations:
b

p + p cos(ln t)+ q sin(ln t) f t, p cos(ln t)+ q sin(ln t) dt =0,
a
b

q + q cos(ln t) – p sin(ln t) f t, p cos(ln t)+ q sin(ln t) dt = A.
a
b

β β
52. y(x)+ y(xt)f t, y(t) dt = Ax cos(ln x)+ Bx sin(ln x).
a
A solution:
β
β
y(x)= px cos(ln x)+ qx sin(ln x), (1)
where p and q are roots of the following system of algebraic (or transcendental) equations:
b

β
β
β

p + t p cos(ln t)+ q sin(ln t) f t, pt cos(ln t)+ qt sin(ln t) dt = A,
a
b (2)

β
β
β

q + t q cos(ln t) – p sin(ln t) f t, pt cos(ln t)+ qt sin(ln t) dt = B.
a
Different solutions of system (2) generate different solutions (1) of the integral equation.
b
β
53. y(x)+ y xt f t, y(t) dt = g(x), β >0.
a
k
1 .For g(x)= n A k x , the equation has a solution of the form
◦
k=1
n
k
y(x)= B k x ,
k=1
where B k are roots of the algebraic (or transcendental) equations
b
n

%
%
kβ
B k + B k F k (B) – A k =0, F k (B)= t f t, B m t m dt.
a m=1
Different roots of this system generate different solutions of the integral equation.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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