(I) Introduction and Application

In this section, we transition from single differential equations to systems of differential equations. We define our variables as follows:

We generally consider systems where the number of equations is equal to the number of dependent variables. For instance, a system with two equations and two dependent variables might look like:

\( f_1(t, x, y, x', y') = 0 \)
\( f_2(t, x, y, x', y') = 0 \)

Application: Coupled Mass-Spring Systems

Consider a system of two masses and two springs. Let \( x(t) \) be the displacement of mass \( m_1 \) from its equilibrium position, and \( y(t) \) be the displacement of mass \( m_2 \) from its equilibrium position. An external force \( f(t) \) is applied to mass \( m_2 \).

A schematic diagram of a mechanical system featuring a fixed wall on the left. A spring labeled K1 connects the wall to a blue rectangular mass labeled m1. A second spring labeled K2 connects mass m1 to a red rectangular mass labeled m2. An external force f(t) is shown as an arrow pushing against mass m2. The displacement labels x(t) and y(t) are positioned below masses m1 and m2 respectively.
Figure: Mechanical System with Two Masses and Two Springs

According to Newton's Second Law, the motion is described by the following system of second-order differential equations:

\( m_1 x'' = -k_1 x + k_2(y - x) \)
\( m_2 y'' = -k_2(y - x) + f(t) \)

(II) First-Order Systems

Any system of differential equations that can be solved for the highest-order derivatives of the dependent variables can be translated into an equivalent system of first-order differential equations. This transformation is the first step for many numerical techniques used with computers.

The Substitution Method

For an \( n^{th} \)-order differential equation \( x^{(n)} = f(t, x, x', \dots, x^{(n-1)}) \), we introduce \( n \) new variables:

\( x_1 = x, \quad x_2 = x', \quad x_3 = x'', \quad \dots, \quad x_n = x^{(n-1)} \)

This yields a system of \( n \) first-order equations:

\( x_1' = x_2 \)
\( x_2' = x_3 \)
\( \vdots \)
\( x_{n-1}' = x_n \)
\( x_n' = f(t, x_1, x_2, \dots, x_n) \)
Example 1

Transform the following third-order differential equation into an equivalent system of first-order differential equations:

\( x^{(3)} + 3x'' + 2x' - 5x = \sin(2t) \)

Solution: First, rewrite the equation solving for the highest derivative:
\( x^{(3)} = 5x - 2x' - 3x'' + \sin(2t) \)

Use the following substitutions:
\( x_1 = x, \quad x_2 = x', \quad x_3 = x'' \)

This yields the system:
\( x_1' = x_2 \)
\( x_2' = x_3 \)
\( x_3' = 5x_1 - 2x_2 - 3x_3 + \sin(2t) \)

Example 2

Transform the following system of second-order equations into an equivalent system of first-order differential equations:

\( 2x'' = -6x + 2y \implies x'' = -3x + y \)
\( y'' = 2x - 2y + 40 \sin(3t) \)

Solution: Introduce four new variables:
\( x_1 = x, \quad x_2 = x', \quad y_1 = y, \quad y_2 = y' \)

This results in the first-order system:
\( x_1' = x_2 \)
\( x_2' = -3x_1 + y_1 \)
\( y_1' = y_2 \)
\( y_2' = 2x_1 - 2y_1 + 40 \sin(3t) \)

To transform a system of \( n \) first-order equations back into a single \( n^{th} \)-order equation, we use successive differentiation of the primary variable and substitute the other system equations to eliminate auxiliary variables.

Example 3

Given the following system of three first-order differential equations, find the equivalent single third-order linear differential equation for the variable \( x_1 \):

(1) \( x_1' = x_2 \)
(2) \( x_2' = x_3 \)
(3) \( x_3' = e^{2t} - 4x_1 + 3x_2 - 2x_3 \)

Derivation:

  1. From (1) and (2): \( x_1'' = x_2' = x_3 \)
  2. Differentiating again: \( x_1''' = x_3' \)
  3. Substitute \( x_2 = x_1' \), \( x_3 = x_1'' \) and \(x_3'= x_1''' \):
    \( x_1''' = e^{2t} - 4x_1 + 3x_1' - 2x_1'' \)

Standard Form:

\[ x_1''' + 2x_1'' - 3x_1' + 4x_1 = e^{2t} \]

Result: This is the equivalent third-order linear differential equation.

(III) Simple 2-Dimensional Systems

Consider the linear second-order differential equation \( x'' + px' + qx = 0 \). By letting \( x = x \) and \( y = x' \), we can transform it into an equavalent two-dimensional system:

\( x' = y \)
\( y' = -py - qx \)
Example 4

Solve the two-dimensional system:
\( x' = -2y \)
\( y' = \frac{1}{2}x \)

Step 1: Convert to a second-order equation.
Differentiate the first equation: \( x'' = -2y' \).
Substitute the second equation into this: \( x'' = -2(\frac{1}{2}x) = -x \).
We have \( x'' + x = 0 \).

Step 2: Solve the second-order equation.
The characteristic equation is \( r^2 + 1 = 0 \implies r = \pm i \).
The general solution is \( x(t) = A \cos t + B \sin t \).

Step 3: Solve for \( y(t) \).
Since \( y = -\frac{1}{2}x' \), we calculate \( x'(t) = -A \sin t + B \cos t \).
Thus, \( y(t) = \frac{1}{2}A \sin t - \frac{1}{2}B \cos t \).

Step 4: Find the particular solution for \( x(0) = 2, y(0) = 0 \).
\( x(0) = A = 2 \).
\( y(0) = -\frac{1}{2}B = 0 \implies B = 0 \).
The solution is \( x(t) = 2 \cos t, y(t) = \sin t \).

Step 5: Identify the trajectory.
Using \( (\frac{x}{2})^2 + y^2 = \cos^2 t + \sin^2 t = 1 \).
This represents an ellipse in the \( xy \)-plane.

Definition: Phase Plane Portrait. A solution \( (x(t), y(t)) \) of a two-dimensional system may be regarded as a parametrization of a solution curve or trajectory in the \( xy \)-plane. The choice of initial conditions determines which trajectory a particular solution parametrizes. A collection of these trajectories is called a phase plane portrait.

A phase plane portrait plot showing multiple concentric blue ellipses centered at the origin of an xy-coordinate system. One particular ellipse is highlighted in red, passing through the point (2,0) on the x-axis and (0,1) on the y-axis, representing the solution x(t) = 2 cos(t) and y(t) = sin(t). All trajectories have arrows indicating a counter-clockwise direction of motion.
Figure: Phase Plane Portrait for the System x' = -2y, y' = x

(IV) Linear Systems

A linear first-order system with \( n \) equations has the following general form:

\( x_1' = p_{11}(t)x_1 + p_{12}(t)x_2 + \dots + p_{1n}(t)x_n + f_1(t) \)
\( x_2' = p_{21}(t)x_1 + p_{22}(t)x_2 + \dots + p_{2n}(t)x_n + f_2(t) \)
\( \vdots \)
\( x_n' = p_{n1}(t)x_1 + p_{n2}(t)x_2 + \dots + p_{nn}(t)x_n + f_n(t) \)

The system is homogeneous if \( f_i(t) = 0 \) for all \( i \); otherwise, it is nonhomogeneous. A solution is an \( n \)-tuple of functions \( x_1(t), \dots, x_n(t) \) that satisfy each equation on an interval \( I \).

Theorem: Existence and Uniqueness for Linear Systems

Suppose that the coefficient functions \( p_{ij} \) and the functions \( f_i \) are continuous on an open interval \( I \) containing the point \( a \). Then the linear system with initial conditions \( x_i(a) = b_i \) for \( 1 \le i \le n \) has a unique solution on the interval \( I \).