This equation is a linear differential equation for 6x, and there 
é ; 7 ‘ F 5 ; 
are standard ways for solving linear differential equations. One first 
considers the linear homogeneous equation 
yay (1.10) 
where in our case y represents the vector 6x and A the matrix Se Let 
yi(t) = Cis Der eae an a (ene 
be the solution of Equation (1.10) with Bee (Ee T) = OEP the Kronecker 
delta; moreover, let @(t, tT) be the matrix whose column vectors are the 
vectors ee O(t, T) = Bes T). The matrix ®(t, T) is called the 
transport matrix or fundamental matrix for the differential Equation 
(1.10). From (1.10) it follows that as a function of t 
= Ge. ©) Shite, © (1.11) 
and by its definition 
OE, t) = I Gea) 
where I is the unit matrix. The solution y(t) is given in terms of its 
value at t = T by 
SAGE) = OCES a) evar) Giesiss)) 
oeeddiae tod E. A. and N. Levinson, "Theory of Ordinary Differential 
Equations," McGraw-Hill Book Company, Inc., New York (1955). 
