Page 437 - Handbook Of Integral Equations
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b
55. y(x)+ y(x – t)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 (2)
b
q + (q – pt)f(t, pt + q) dt – B =0.
a
Different solutions of system (2) generate different solutions (1) of the integral equation.
b
λx
56. y(x)+ y(x – t)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 –λt
k + kF(k) – A =0, F(k)= f t, ke e dt.
a
b
57. y(x)+ y(x – t)f t, y(t) dt = A sinh λx.
a
A solution:
y(x)= p sinh λx + q cosh λx, (1)
where p and q are roots of the following system of algebraic (or transcendental) equations:
b
p + (p cosh λt – q sinh λt)f t, p sinh λt + q cosh λt dt = A,
a (2)
b
q + (q cosh λt – p sinh λt)f t, p sinh λt + q cosh λt dt =0.
a
Different solutions of system (2) generate different solutions (1) of the integral equation.
b
58. y(x)+ y(x – t)f t, y(t) dt = A cosh λx.
a
A solution:
y(x)= p sinh λx + q cosh λx,
where p and q are roots of the following system of algebraic (or transcendental) equations:
b
p + (p cosh λt – q sinh λt)f t, p sinh λt + q cosh λt dt =0,
a
b
q + (q cosh λt – p sinh λt)f t, p sinh λt + q cosh λt dt = A.
a
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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