322 Ordinary Differential Equations
DEFINITION Linear System Initial Value Problem
A linear system initial value problem consists of solving the system
of n first-order equ ations(14a)subject to the n initial conditions(14b)Calculating the partial derivatives of fi with respect to yj in the linear
system (14a), we find 8 fd ayj = aij (x). If t he n^2 + n functions aij (x), i, j =
1, 2,.. ., n and bi (x), i = 1, 2,.. ., n a re all defined and continuous on some
interval I = (a, /3) which contains c, t hen all of the functions fi and afif ayj
will b e defined and continuous functions of x, y 1 , y 2 , ... , Yn on the generalized
rectangle
R = {( x, Y1, Y2, ... , Yn) I et < x < /3 and -oo < Yi < oo for i = 1, 2, .. ., n}.
Therefore, by the fundamental existence and uniqueness theorem and by the
continuation theorem for system initial value problems, there exists a unique
solution to the initial value problem (14) on t h e interval I= (a, /3). Hence, we
h ave the following fundamental existence and uniqueness theorem for linear
system initial value problems.
FUNDAMENTAL EXISTENCE AND UNIQUENESS THEOREM
FOR LINEAR SYSTEM INITIAL VALUE PROBLEMSIf the functions aij(x), i , j = 1,2, ... , n and bi(x), i = 1, 2, .. .,n are all
defined and continuous on the interval I = (a , /3) and if c E J , then there
exists a unique solution to the linear system initial value problem (14) on
the interval I.