First Order Systems
Consider the motion of a particle in three-dimensional space.
To describe
this using Newton's laws of motion, we would use a system of differential
equations, one for each coordinate (say , , ). On the left
is the mass times a component of the acceleration, and on the right
is the corresponding component of the force, which may depend on the
position and velocity of the particle as well as the time
(which we are using as independent variable).
This would be a second order system. We want to discuss first order
systems. We can make our system into a first order system
by using the three components of the velocity as
additional dependent variables. We then get six equations
in six dependent variables:
In general, higher order systems can be reduced to first order
systems (at the cost of increasing the number of variables and equations).
For example, a third order equation
can be written
as a first order system using variables , , :
Therefore we will concentrate on first order systems.
We can write a first order system in general as
To determine the motion of our particle moving in space, we would specify
as initial conditions the three components of position and the three components
of velocity at some particular time. In general, for a first order system,
a set of initial conditions will specify the values of all the dependent
variables at some particular :
,
, ...
.
First order systems have an existence and uniqueness theorem very
much like that of first order equations.
Theorem: Consider a system (as above) where the functions
and their partial derivatives
are all continuous
on a rectangular region ,
. Take any
initial conditions where and
for all .
Then there is a unique solution of the initial value problem consisting
of the system and initial conditions, defined for in some interval
around . At the endpoints of this interval the solution reaches the
boundary of our region (either or , or some or ).
As in the case of single equations, more can be said if the system is linear.
A linear system is of the form
Suppose the functions
and are all continuous on
an interval (with and/or allowed).
Then in the conclusion of the Existence and Uniqueness Theorem, the
unique solution is defined for . Thus while
solutions of a nonlinear system may go off to at some finite
value of , even though the equation seems perfectly well-behaved
there, solutions of a linear system can not do this.
Linear systems are very important, and we'll spend this section of the
course
investigating them.
Then in the next section we'll come back to nonlinear
systems, using linear systems as one of our main tools for investigating
them.
It is convenient to use the notation of matrices and vectors instead of
writing out the systems in full. We put the dependent variables
, , ... together into a vector, written as a column
and the nonhomogeneous
terms , , ... into another vector
The
coefficients go into a matrix with rows and columns
(an matrix)
Then
we write the system of equations as
.
Note that our vectors and matrices have entries that may be functions of
the independent variable , and the derivative is the vector whose
entries are the derivatives of the entries of . In all the examples
we'll actually do,
will be a constant matrix. The is an example of the multiplication
of a matrix and a vector. The rule for this is:
If is an matrix with entries and is an
-component vector with entries , then is the -component vector
Structure of Solutions
Just as for second-order linear equations, linear algebra tells us some
important facts about the structure of the solutions of linear systems.
- Superposition (homogeneous):
For any two solutions
and
of the homogeneous system
and any constant numbers
and ,
is also a solution.
- Superposition (nonhomogeneous): If
and
are solutions of the nonhomogeneous
systems
and
respectively, and and are constant numbers, then
is a solution of the system
.
- General solution (homogeneous): The general solution of the homogeneous system
is
where
are arbitrary
constant numbers and
are a fundamental set of
solutions. A fundamental set of solutions consists of solutions
such that for some is in the interval where
is continuous, every vector with components can be represented as a
linear combination
.
This means that we can match any initial condition at .
The theory then
says that this is true for all in the interval.
- General solution (nonhomogeneous): If is one solution of the nonhomogeneous system
and
are a fundamental set of solutions of the
homogeneous system
, then the general solution
of the nonhomogeneous system
is
where
are arbitrary
constant numbers.
For example, the homogeneous system
happens to have the following three solutions (don't worry about how I found them):
For these vectors are
It's easy to get a solution of the form
to satisfy any initial condition at
: for the initial condition
you just take , , .
For an initial condition at some other , say
, it's a bit more work, as
you'd have to solve three equations for the three unknowns , ,
:
Robert Israel
2002-03-12