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n m–1
9.2-4. Equations With Kernel of the Form K(x, t)= ϕ m (t)(t – x)
m=1
Let us represent the resolvent of this degenerate kernel in the form
n
d v(x, t)
(n) (n)
R(x, t)= –v t (x, t), v t = ,
dt n
where the auxiliary function v(x, t) vanishes at t = x together with n–2 derivatives with respect to t,
and the (n – 1)st derivative with respect to t at t = x is equal to 1. On substituting the expression for
the resolvent into Eq. (3) of Subsection 9.1-1, we obtain
x
(n) (n)
v t (x, t)= K(s, t)v s (x, s) ds – K(x, t).
t
Let us apply integration by parts to the integral on the right-hand side. Taking into account the
properties of the auxiliary function v(x, t), we arrive at the following Cauchy problem for an
nth-order ordinary differential equation:
v (n) + ϕ 1 (t)v (n–1) + ϕ 2 (t)v (n–2) +2ϕ 3 (t)v (n–3) + ··· +(n – 1)! ϕ n (t)v =0,
t t t t
(n–2) (n–1)
v = v = ··· = v =0, v =1.
t=x t t=x t t=x t t=x
The parameter x occurs only in the initial conditions, and the equation itself is independent of x
explicitly.
Remark 4. A kernel of the form K(x, t)= n φ m (t)x m–1 can be reduced to a kernel of the
m=1
above type by elementary transformations.

9.2-5. Equations With Degenerate Kernel of the General Form
In this case, the Volterra equation of the second kind can be represented in the form
n x

y(x) – g m (x) h m (t)y(t) dt = f(x). (14)
a
m=1
Let us introduce the notation
x

w j (x)= h j (t)y(t) dt, j =1, ... , n, (15)
a
and rewrite Eq. (14) as follows:
n

y(x)= g m (x)w m (x)+ f(x). (16)
m=1
On differentiating the expressions (15) with regard to formula (16), we arrive at the following system
of linear differential equations for the functions w j = w j (x):
n

w = h j (x) g m (x)w m + f(x) , j =1, ... , n,
j
m=1
with the initial conditions
w j (a)=0, j =1, ... , n.
Once the solution of this system is found, the solution of the original integral equation (14) is defined
by formula(16) or any of the expressions
w (x)

j
y(x)= , j =1, ... , n,
h j (x)
which can be obtained from formula (15) by differentiation.
•
References for Section 9.2: E. Goursat (1923), H. M. M¨ untz (1934), A. F. Verlan’ and V. S. Sizikov (1986), A. D. Polyanin
and A. V. Manzhirov (1998).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 463