Page 129 - Handbook Of Integral Equations

2 . Solution with λ <0:
◦
x
√
y(x)= f(x)+ k sinh[k(x – t)]f(t) dt, k = –λ.
a
x

5. y(x)+ A + B(x – t) y(t) dt = f(x).
a
2
1 . Solution with A >4B:
◦
x

y(x)= f(x) – R(x – t)f(t) dt,
a
2
2B – A
1 1 2
R(x)=exp – Ax A cosh(βx)+ sinh(βx) , β = A – B.
2 2β 4
2
◦
2 . Solution with A <4B:
x

y(x)= f(x) – R(x – t)f(t) dt,
a
2
2B – A
1 1 2
R(x)=exp – Ax A cos(βx)+ sin(βx) , β = B – A .
2 2β 4
2
◦
3 . Solution with A =4B:
x
1 1 2
y(x)= f(x) – R(x – t)f(t) dt, R(x)=exp – Ax A – A x .
2 4
a
x

6. y(x) – Ax + Bt + C y(t) dt = f(x).
a
For B = –A see equation 2.1.5. This is a special case of equation 2.9.6 with g(x)= –Ax and
h(t)= –Bt – C.
x
By differentiation followed by the substitution Y (x)= y(t) dt, the original equation
a
can be reduced to the second-order linear ordinary differential equation

Y xx – (A + B)x + C Y – AY = f (x) (1)
x
x
under the initial conditions

Y (a)=0, Y (a)= f(a). (2)
x
A fundamental system of solutions of the homogeneous equation (1) with f ≡ 0 has the
form
1 2 1 2
Y 1 (x)= Φ α, ; kz , Y 2 (x)= Ψ α, ; kz ,
2 2
A A + B C
α = , k = , z = x + ,
2(A + B) 2 A + B

where Φ α, β; x and Ψ α, β; x are degenerate hypergeometric functions.
Solving the homogeneous equation (1) under conditions (2) for an arbitrary function

f = f(x) and taking into account the relation y(x)= Y (x), we thus obtain the solution of the
x
integral equation in the form
x

y(x)= f(x) – R(x, t)f(t) dt,
a
√
∂ 2 Y 1 (x)Y 2 (t) – Y 2 (x)Y 1 (t) 2 πk
C 2

R(x, t)= , W(t)= exp k t + .
∂x∂t W(t) Γ(α) A + B
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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