AN INTEGRAL EQUATION ASSOCIATED WITH LINEAR HOMOGENEOUS DIFFERENTIAL EQfJATIONS

Associated with each linear homogeneous differential equation y(n)=∑i=0n−1ai(x)y(i) of order n on the real line, there is an equivalent integral equation f(x)=f(x0)


n-I (n)
(i) y Z ai(x)y of order n on the real line, there is an equivalent integral which is satisfied by each solution f(x) of the differential equation.
KEY WORDS AND PHRASES.Linear homogeneous differential equations, Integral equations, Initial value problems, Variation of parameters formula, Uniform convergence.Let n be a positive integer, I be an interval on the real line R and C(1) be the class of all functions continuous on I. Let n-I be any n-th order (normalized) ordinary linear homogeneous differential equation, where a i(x) e C(I), i=0,I,2 (n-l).
The purpose of this article is to derive an equivalent integral equation satisfied by the solutions of the linear homogeneous differential equation (I.I).
THEOREM.Let f(x) be a solution of (l.l) defined on I and x 0 e I. Then f(x)   is also a solution of the integral equation where h(x) is the unique solution of the (n-l)-th order linear homogeneous differential equation and G (x,u) is the well-known Green's Function associated with the homogeneous n-I equation (2.2) PROOF OF THE THEOREM.In order to deduce the integral equation ( 2. I), we will use the well-known Variation of Parameters formula X y(x) Consider the sequence of functions: Clearly, for each k, fk(x0) f(Xo) fk(x) is differentiable on I and for k 2 X fk(x) h(x) + I Gn_l(X,u)a0(u)fk_l(U)du. (2.5) x 0 Using (2.3), we conclude that, for each k a 2, fk(x) is the unique solution of the non-homogeneous initial value problem Ifk+ l(x) fk(x) -< Mksk(b-a)2k/2k!Ifk+ l(x) fk(x) _< Mksk(b-a)k/k!using recursively the bounds for fi+ l(x) fi(x) i 1,2,3, k-I k= for all x e I.In particular g(x O) f(Xo).
Also from (2.5) we get by taking limit as k- f(Xo) + I h(u)du + I [ Gn_l(U,V)ao(v)g(v)dv]du x 0 x 0 x 0 Again, relation (2.7) implies by (2.3) that g (x) is the unique solution of the initial value problem (2.7) (2.8) Y(n-l) an_l(x)y(n-2) + + a l(x)y + aO(x)g(x Hence f(x) g(x) for all x e I. Therefore, by (2.8) This completes the proof.
REMARK.The above proof clearly shows how a solution of a linear homogeneous equation with prescribed initial values can be constructed out Qf a solution h(x) and the Green's Function G (x u) of a lower order homogeneous linear equation.n-I This is specially significant in case of second order homogeneous equations, as A(x)-A(u) solutions ceA-X't and the Green's function Gl(X,U) e x [A(x) f a l(u)du], of first order homogeneous equation y a l(x)y are readily x 0 available.
fk(x) e (I).Both the sequences {fk(x)} and {fk(x)} converge uniformly on every compact subset on the interval I. To see this, let B be a compact subset of I. Then there exists a closed and bounded interval[a,b]  such that B[a,b]    l(U'v)a0(v)l s max Ifl(v) for each u,v[a,b].One can now see very easily that for each x e [a,b], Since each of the series Mksk(b-a)2k/2k!, Mksk(b-a)k/k!converges, we conclude by Weierstrass' M-test that each of the series of functions fl(x) + (fk+l (x) fk(x)), fl(x) + (fk+l (x) fk(x)) k=1 k=l converges uniformly on [a,b] and hence on B. Therefore there is a function g(x) e C I(I) such that fl (x) + I (fk+l (x) fk (x)) lira fk(x) g(x) k=l k+= fl(x) + .(fk+l(X) fk(x)) lira fk(x) g (x)