Systems of Differential Equations

1 Matrices and Systems of Linear Equations

An n × m matrix is an array A = (a_ij) of the form

where each a_ij is a real or complex number.
The matrix has n rows and m columns.
For 1≤j≤m, 1≤i≤n, the n × 1 matrix

is called the j−th column of A and the 1×m matrix

is called the i − th row of A.
We can add n × m matrices as follows. If A = (a_ij)
and B = (b_ij), then C = A+B is the matrix (c_ij) defined
by

We can multiply an n × m matrix A = (a_ij) by an
m × p matrix B_jk to get an n × p matrix C = (c_ik)
defined by

Thus the element c_ik is the dot product of the ith row
of A and the jth column of B.
Both the operations of matrix addition and matrix
multiplication are associative. That is,

(A + B) + C = A + (B + C), (AB)C = A(BC).

Multiplication of n×n matrices is not always commutative.
For instance, if

then,

We will write vectors x = (x₁, . . . , x_n) in Rⁿ both as
row vectors and column vectors.
Matrices are useful for dealing with systems of linear
equations.

Since our interest here is in treating systems of differential
equations, we will only consider linear systems of
n equations in n unknowns.
We can write the system

as a single vector equation
Ax = b

where A in the n × n matrix (a_ij), x is an unknown
n−vector, and b is a known n−vector.
Let e_i be the n−vector with zeroes everywhere except
in the i−th position and a 1 there.
The n×n matrix I whose i−th row is e_i is called the
n × n identity matrix.
For any n × n matrix A we have

AI = IA = A.

An n × n matrix A is invertible if there is another
n × n matrix B such that AB = BA = I. We also
call A non-singular. A singular matrix is one that is not
invertible.

The matrix B is unique and called the inverse of A.
It is usually written A⁻¹.
Let 0 denote the n−vector all of whose entries are 0.
A collection u₁, u₂, . . . , u_k of vectors in Rⁿ is called a
linearly independent set of vectors in Rⁿ, if whenever
we have a linear combination

with the constants (scalars) we must have
for every i.

Ax = b

Fact. The following conditions are equivalent for an
n × n matrix A to be invertible.

1. the rows of A form a linearly independent set of vectors
2. the columns of A form a linearly independent set of
vectors
3. for every vector b, the system
Ax = b
has a unique solution
4. det(A) ≠ 0

We define the number det(A) inductively by

where A[i | 1] is the (n−1)×(n−1) matrix obtained
by deleting the first column and i−th row of A.
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2 Systems of Differential Equations

Let U be an open subset of Rⁿ, I be an open interval in
R and : I×Rⁿ -> Rⁿ be a function from I ×Rⁿ to Rⁿ.
The equation

is called a first order ordinary differential equation in
Rⁿ. We emphasize here that x is an n−dimensional vector
in Rⁿ. We also consider the initial value problem

where t₀ ∈I and x₀ ∈U.
A solution to the IVP (2) is a differentiable function
x(t) from an open subinterval containing t₀ such
that

for t ∈J.
The general solution to (1) is an expression

x(t, c) (3)

where c is an n−dimensional constant vector in Rⁿ
such that every solution of (1) can be written in the form
(3) for some choice of c.

If we write out the D.E. (1) in coordinates, we get a
system of first order differential equations as follows.

Fact: The n−th order scalar D.E. is equivalent to a
simple n−dimensional system.

Consider

Letting we get

If we have a solution y(t) to (5), and set x₁ = y(t), x₂(t) =
y'(t), . . . , x_n(t) = y⁽ⁿ⁻¹⁾(t), then x(t) = (x₁(t), . . . , x_n(t)
is a solution to the system (6). Conversely, if we have a
solution x(t) = (x₁(t), . . . , x_n(t) to the system 6, then
putting y(t) = x₁(t) gives a solution to (5).

The following existence and uniqueness theorem is proved
in more advanced courses.

Theorem (Existence-Uniqueness Theorem for
systems). U be an open set in Rⁿ, and let I be an
open interval in R. Let f (t, x) be a C¹ function of the
variables (t, x) defined in I × U with values in Rⁿ.
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
on time, then one calls the system autonomous
(or time-independent). Otherwise, one calls the system
non-autonomous or time-dependent.
There is a simple geometric description of autonomous
systems in Rⁿ. In that case, we consider

where f is a C¹ function defined in an open subset U
in Rⁿ. We think of f as a vector field in U and solutions
x(t) of (7) as curves in U which are everywhere tangent
to f .

2.1 Linear Systems of Differential Equations: General
Properties

The system

in which A(t) is a continuous n × n matrix valued
function of t and g(t) is a continuous n−vector valued
function of t is called a linear system of differential equations
(or a linear differential equation) in Rⁿ.

As in the case of scalar equations, one gets the general
solution to (8) in two steps. First, one finds the general
solution x_h(t) to the associated homogeneous system

Then, one finds a particular solution x_p(t) to (8) and
gets the general solution to (8) as a sum

Accordingly, we will examine ways of doing both tasks.

Let y_i(t) be a collection of Rⁿ−valued functions for
1≤i≤k. We say that they form a linearly independent
set of functions if whenever are k scalars
such that

for all t, we have that
An n × n matrix of linearly independent solutions
to the homogeneous linear system (9) is called a
fundamental matrix for (9).

A necessary and sufficient condition for the matrix
of solutions to be a fundamental matrix is that
det() = 0 for some (or any ) t.

If y₁(t), . . . , y_n(t) are n solutions to (9), and is
the matrix whose columns are the functions y_i(t), then
the function

is called the Wronskian of the collection {y₁(t), . . . , y_n(t)}
of solutions. It is then a fact that W(t) vanishes at some
point t₀ if and only if it vanishes at all point t.

The general solution to (9) has the form

where is any fundamental matrix for (9) and c is
a constant vector.

Thus, we have to find fundamental matrices and particular
solutions. We will do this explicitly below for
n = 2, 3 and constant matrices A.

To close this section, we observe an analogy between
systems

and the scalar equation x' = ax.
One can define the matrix exp(A) = e^A by the power
series

It can be shown that the matrix series on the right
side of this equation converges for any A. The series
represents a matrix with many properties analogous to
the usual exponential function of a real variable.

In particular, for a real number t, the matrix function
is a differentiable matrix valued funtion and its
derivative, computed by differentiating the series

term by term satisfies

It follows that, for each vector x₀, the vector function

is the unique solution to the IVP

Hence, the matrix e^tA is a fundamental matrix for the
system

This observation is useful in certain circumstances,
but, in general, it is hard to compute e^tA directly. In
practice the methods involving eigenvalues described in
the next secion are easier to use to find the general solution
to