8.6. The Dimension Of The Solution Space Of A Homogeneous Linear
be a linear differential operator of order n given by
Then the solution space of the homogeneous
has dimension n.
be the linear transformation that maps each
function f in the solution space
to the initial- value vector of f at
. By the uniqueness theorem,
. Hence, by Theorem 4.10, T is 1-1 on
exists and is also 1-1. Furthermore, since every element in Rn
can be an initial-value vector, the map is onto. By Theorem 4.11,
be a linear differential operator of order n.
solutions of the homogeneous equation
on J. Then every solution
can be expressed as a linear combination
This is a corollary of Theorem 8.4.
Eq(8.8) is called a general solution of
The dimensionality theorem asserts that an nth order LDE always has n independent
solutions. There is however no general
method for obtaining these solutions, except for a few special types of
equations, e.g., those with constant coefficients.