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## Systems of Differential Equations## 1 Matrices and Systems of Linear EquationsAn n × m matrix is an array A = (a
where each a
is called the j−th column of A and the 1×m matrix
is called the i − th row of A.
We can multiply an n × m matrix A = (a
Thus the element c (A + B) + C = A + (B + C), (AB)C = A(BC). Multiplication of n×n matrices is not always commutative.
then,
We will write vectors x = (x Since our interest here is in treating systems of differential
as a single vector equation where A in the n × n matrix (a AI = IA = A. An n × n matrix A is invertible if there is another The matrix B is unique and called the inverse of A.
with the
constants (scalars) we must have
1. the rows of A form a linearly independent set of vectors We define the number det(A) inductively by
where A[i | 1] is the (n−1)×(n−1) matrix obtained ## 2 Systems of Differential EquationsLet U be an open subset of R
is called a first order ordinary differential equation in
where t
for t ∈J. x(t, c) (3) where c is an n−dimensional constant vector in R If we write out the D.E. (1) in coordinates, we get a
Consider
Letting we get
If we have a solution y(t) to (5), and set x The following existence and uniqueness theorem is proved
^{n}, and let I be anopen interval in R. Let f (t, x) be a C ^{1} function of thevariables (t, x) defined in I × U with values in R ^{n}.Then, there is a unique solution x(t) to the initial value problem
If the right side of the system f (t, x) does not depend
where f is a C
The system
in which A(t) is a continuous n × n matrix valued As in the case of scalar equations, one gets the general
Then, one finds a particular solution x
Accordingly, we will examine ways of doing both tasks. Let y
for all t, we have that
A necessary and sufficient condition for the matrix If y
is called the Wronskian of the collection {y The general solution to (9) has the form
where
is any fundamental matrix for (9) and c is Thus, we have to find fundamental matrices and particular To close this section, we observe an analogy between
and the scalar equation x' = ax.
It can be shown that the matrix series on the right In particular, for a real number t, the matrix function
term by term satisfies
It follows that, for each vector x
is the unique solution to the IVP
Hence, the matrix e This observation is useful in certain circumstances, |