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 $x_1$, $x_2$, $x_3$). On the left is the mass $m$ 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 $t$ (which we are using as independent variable).

\begin{displaymath}
\begin{array}{ll}
m x''_1&= F_1(t, x_1, x_2, x_3, x'_1, x'_...
...
m x''_3&= F_3(t, x_1, x_2, x_3, x'_1, x'_2, x'_3)
\end{array}\end{displaymath}

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:

\begin{displaymath}
\begin{array}{ll}
x_1' &= v_1 \qquad\qquad v_1' = F_1(t, x...
...\qquad v_3' = F_3(t, x_1, x_2, x_3, v_1, v_2, v_3)
\end{array}\end{displaymath}

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 $y''' = f(t, y, y', y'')$ can be written as a first order system using variables $y_1 = y$, $y_2 = y'$, $y_3 = y''$:

\begin{displaymath}y_1' = y_2, \qquad y_2' = y_3,\qquad y_3' = f(t, y_1, y_2, y_3)\end{displaymath}

Therefore we will concentrate on first order systems. We can write a first order system in general as

\begin{displaymath}
\begin{array}{ll}
x_1' &= F_1(t, x_1, x_2, \ldots, x_n)\\ ...
...\ldots \\
x_n' &= F_n(t, x_1, x_2, \ldots, x_n)
\end{array}\end{displaymath}

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 $t$: $x_1(t_0) = x_1^{0}$, $x_2(t_0) = x_2^0$, ... $x_n(t_0) = x_n^0$. 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 $F_i$ and their partial derivatives $\partial F_i/\partial x_j$ are all continuous on a rectangular region $a \le t \le b$, $a_i \le x_i \le b_i$. Take any initial conditions where $a < t_0 < b$ and $a_i < x_i^0 < b_i$ for all $i$. Then there is a unique solution of the initial value problem consisting of the system and initial conditions, defined for $t$ in some interval around $t_0$. At the endpoints of this interval the solution reaches the boundary of our region (either $t = a$ or $b$, or some $x_i = a_i$ or $b_i$). As in the case of single equations, more can be said if the system is linear. A linear system is of the form

\begin{displaymath}\begin{array}{rl}
x_1' &= p_{11}(t) x_1 + p_{12}(t) x_2 + \l...
...+ p_{n2}(t) x_2 + \ldots + p_{nn}(t) x_n + g_n(t)
\end{array}\end{displaymath}

Suppose the functions $\displaystyle p_{ij}(t)$ and $g_i(t)$ are all continuous on an interval $a < t < b$ (with $a=-\infty$ and/or $b=+\infty$ allowed). Then in the conclusion of the Existence and Uniqueness Theorem, the unique solution is defined for $a < t < b$. Thus while solutions of a nonlinear system may go off to $\infty$ at some finite value of $t$, 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 $x_1$, $x_2$, ...$x_n$ together into a vector, written as a column

\begin{displaymath}{\bf x}= \pmatrix{x_1\cr x_2\cr ...\cr x_n\cr}
\end{displaymath}

and the nonhomogeneous terms $g_1(t)$, $g_2(t)$, ...$g_n(t)$ into another vector

\begin{displaymath}{\bf g}(t) = \pmatrix{g_1(t)\cr g_2(t)\cr ...\cr g_n(t)\cr}
\end{displaymath}

The coefficients $p_{ij}(t)$ go into a matrix with $n$ rows and $n$ columns (an $n \times n$ matrix)

\begin{displaymath}P = \pmatrix{ p_{11} & p_{12} & \ldots & p_{1n}\cr
p_{21} & p...
...s & \ldots & & \ldots\cr
p_{n1} & p_{n2} & \ldots & p_{nn}\cr}
\end{displaymath}

Then we write the system of equations as ${\bf x}' = P {\bf x}+ {\bf g}(t)$. Note that our vectors and matrices have entries that may be functions of the independent variable $t$, and the derivative ${\bf x}'$ is the vector whose entries are the derivatives of the entries of ${\bf x}$. In all the examples we'll actually do, $P$ will be a constant matrix. The $P {\bf x}$ is an example of the multiplication of a matrix and a vector. The rule for this is:

If $P$ is an $m \times n$ matrix with entries $p_{ij}$ and ${\bf x}$ is an $n$-component vector with entries $x_j$, then $P {\bf x}$ is the $m$-component vector

\begin{displaymath}\pmatrix{ p_{11} x_1 + p_{12} x_2 + \ldots + p_{1n} x_n \cr
...
...\ldots \cr
p_{m1} x_1 + p_{m2} x_2 + \ldots + p_{mn} x_n \cr} \end{displaymath}

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.
  1. Superposition (homogeneous): For any two solutions $\displaystyle {\bf x}^{(1)}$ and $\displaystyle {\bf x}^{(2)}$ of the homogeneous system ${\bf x}' = P(t) {\bf x}$ and any constant numbers $c_1$ and $c_2$, $\displaystyle c_1 {\bf x}^{(1)} + c_2 {\bf x}^{(2)}$ is also a solution.
  2. Superposition (nonhomogeneous): If $\displaystyle {\bf x}^{(1)}$ and $\displaystyle {\bf x}^{(2)}$ are solutions of the nonhomogeneous systems ${\bf x}' = P(t) {\bf x}+ {\bf g}_1(t)$ and ${\bf x}' = P(t) {\bf x}+ {\bf g}_2(t)$ respectively, and $c_1$ and $c_2$ are constant numbers, then $c_1 {\bf x}^{(1)} + c_2 {\bf x}^{(2)}$ is a solution of the system $\displaystyle {\bf x}' = P(t) {\bf x}+ c_1 {\bf g}_1(t) + c_2 {\bf g}_2(t)$.
  3. General solution (homogeneous): The general solution of the $n \times n$ homogeneous system ${\bf x}' = P(t) {\bf x}$ is $\displaystyle {\bf x}= c_1 {\bf x}^{(1)} + ... + c_n {\bf x}^{(n)}$ where $c_1, \ldots, c_n$ are arbitrary constant numbers and $\displaystyle {\bf x}^{(1)}, \ldots, {\bf x}^{(n)}$ are a fundamental set of solutions. A fundamental set of solutions consists of $n$ solutions $\displaystyle {\bf x}^{(1)}, \ldots, {\bf x}^{(n)}$ such that for some $t_0$ is in the interval where $P(t)$ is continuous, every vector with $n$ components can be represented as a linear combination $\displaystyle c_1 {\bf x}^{(1)}(t_0) + \ldots + c_n {\bf x}^{(n)}(t_0)$. This means that we can match any initial condition at $t=t_0$. The theory then says that this is true for all $t_0$ in the interval.
  4. General solution (nonhomogeneous): If ${\bf x}^{(p)}$ is one solution of the nonhomogeneous system ${\bf x}' = P(t) {\bf x}+ {\bf g}(t)$ and ${\bf x}^{(1)}, \ldots, {\bf x}^{(n)}$ are a fundamental set of solutions of the homogeneous system ${\bf x}' = P(t) {\bf x}$, then the general solution of the nonhomogeneous system ${\bf x}' = P(t) {\bf x}+ {\bf g}(t)$ is ${\bf x}= {\bf x}^{(p)} + c_1 {\bf x}^{(1)} + ... + c_n {\bf x}^{(n)}$ where $c_1, \ldots, c_n$ are arbitrary constant numbers.

For example, the homogeneous system

\begin{displaymath}{\bf x}' = \pmatrix{-5 & -15 & -5\cr
1 & -1 & 3\cr
2 & -2 & 6\cr} {\bf x}\end{displaymath}

happens to have the following three solutions (don't worry about how I found them):

\begin{displaymath}{\bf x}^{(1)}(t) = \pmatrix{1-5t \cr t\cr 2 t\cr} \qquad
{\b...
...\pmatrix{-5 t - 25 t^2 \cr 3 t + 5 t^2\cr 1 + 6 t + 10 t^2\cr}
\end{displaymath}

For $t_0 = 0$ these vectors are

\begin{displaymath}{\bf x}^{(1)}(0) = \pmatrix{ 1 \cr 0 \cr 0\cr}
\qquad {\bf x}...
...\cr 0\cr}
\qquad {\bf x}^{(3)}(0) = \pmatrix{ 0 \cr 0 \cr 1\cr}\end{displaymath}

It's easy to get a solution of the form $ c_1 {\bf x}^{(1)} + c_2 {\bf x}^{(2)} + c_3 {\bf x}^{(3)}$ to satisfy any initial condition at $t = 0$: for the initial condition $\displaystyle {\bf x}(0) = \pmatrix{ a_1\cr a_2 \cr a_3\cr}$ you just take $c_1 = a_1$, $c_2 = a_2$, $c_3 = a_3$. For an initial condition at some other $t$, say $\displaystyle {\bf x}(1) = \pmatrix{ a_1\cr a_2 \cr a_3\cr}$, it's a bit more work, as you'd have to solve three equations for the three unknowns $c_1$, $c_2$, $c_3$:


\begin{displaymath}\begin{array}{rl}
c_1 x^{(1)}_1(1) + c_2 x^{(2)}_1(1) + c_3 x...
..._3(1) + c_2 x^{(2)}_3(1) + c_3 x^{(3)}_3(1) &= a_3 \end{array} \end{displaymath}





Robert Israel
2002-03-12