Page 305 - A Course in Linear Algebra with Applications

8.3: Applications to Systems of Linear Differential Equations 289

Here the Qj<ij £1X6 assumed to be constants. The object is to
find the most general functions 2/i,...,j/ n , differentiable in
some interval [a ,b], which satisfy the equations of the system.
Alternatively one may wish to find functions which satisfy in
addition a set of initial conditions of the form

yi(x Q) = h, y 2(x 0) = b 2, ..., y n(x n) = K-

Here the bi are certain constants and x$ is in the interval [a, b].
Let A = [a,ij], the coefficient matrix of the system and
write
fyi\

Y =

\y n/

Then we define the derivative of Y to be

/y[\
y'2
Y' =

With this notation the given system of differential equations
can be written in matrix form

Y' = AY.

By a solution of this equation we shall mean any column
vector Y of n functions in D[a, b] which satisfies the equation.
The set of all solutions is a subspace of the vector space of all
n-column vectors of differentiable functions; this is called the
solution space. It can be shown that the dimension of the so-
lution space equals n, so that there are n linearly independent
solutions, and every solution is a linear combination of them.