BSU Math 333 (Ultman)
In order to learn the new skills and ideas presented in this worksheet, you must:
Know what a first-order linear differential equation is.
Know how to check whether a given function is the solution of a differential equation.
Be able to write a system of linear (algebraic) equations as a matrix equation.
In this worksheet, you will:
Learn terminology associated with systems of two first order linear differential equations.
Convert systems of first order linear differential equations into matrix equations.
Determine whether a given pair of equations is s solution to a given system of two first order linear differential equations.
Identify critical points (equilibrium solutions).
At the end of this worksheet, we will (as a class) look at a graphical way of analyzing the behavior of solutions to systems of two first order linear differential equations.

BSU Math 333 (Ultman) Worksheet: Introduction to Systems of First Order Linear Differential Equations 1
A system of first order differential equations is a collection of first order differential equations that need to be satisfied simultaneously.
Three examples of systems of first order differential equations are: x
0 y
0
= 5 x + 12 y
= 12 x − 42 y
(1) x
0 y
0
= 3 x − 7 y − 6
= 12 x + y + 10
(2) x
0 y
0
= 0 .
5 x − 0 .
75 y + sin t
= x + 1 .
3 y − 2 cos t
(3)
•
These are all examples of linear systems, since they can be written in the form: x
0
= p
11
( t ) x + p
12
( t ) y + g
1
( t ) y
0
= p
21
( t ) x + p
22
( t ) y + g
2
( t )
Where p
11
( t ) , p
12
( t ) , p
21
( t ) , p
22
( t ) , g
1
( t ) , g
2
( t ) are functions of t .
•
Systems of first order linear differential equations can be expressed in matrix form:
0
= A~ +
~
.
For example, the system: x
0
= p
11
( t ) x + p
12
( t ) y + g
1
( t ) y
0
= p
21
( t ) x + p
22
( t ) y + g
2
( t )
Can be expressed as the matrix equation: x
0 y
0
= p p
11
21
( t ) p
( t ) p
12
22
( t )
( t ) x y
+ g
1
( t ) g
2
( t )
.
•
A system is autonomous if independent variable t does not appear explicitly in the equations. A system is nonautonomous if there are explicit functions of t in the equation.
•
A system is homogenous if the vector
~
=
~
. A system is nonhomogenous if
~
=
~
.

BSU Math 333 (Ultman) Worksheet: Introduction to Systems of First Order Linear Differential Equations 2
1. Express the first system of equations as a matrix equation, determine whether it is autonomous or nonautonomous, and whether it is homogeneous or nonhomogenous.
2. Express the second system of equations as a matrix equation, determine whether it is autonomous or nonautonomous, and whether it is homogeneous or nonhomogenous.
3. Express the third system of equations as a matrix equation, determine whether it is autonomous or nonautonomous, and whether it is homogeneous or nonhomogenous.

BSU Math 333 (Ultman) Worksheet: Introduction to Systems of First Order Linear Differential Equations 3
•
A solution to a system of differential equations is a set of functions that satisfies all equations simultaneously.
4. Convert the system: x
0
= 2 x − y y
0
= 3 x − 2 y into a matrix equation. Determine whether it is autonomous or nonautonomous, and whether it is homogeneous or nonhomogeneous. Then, show that: x
1
= x y
(
( t t
)
)
= e e t t and ~
2
= x y
(
( t t
)
)
= e
− t
3 e
− t
.
are both solutions to the system.

BSU Math 333 (Ultman) Worksheet: Introduction to Systems of First Order Linear Differential Equations 4
•
A critical point (or equilibrium solution to a system of differential equations is a constant vector × such that ~
0
= ~ .
Since x
0
= A~ +
~
, you can find the critical points by solving the system A~ +
~
= 0 , or equivalently:
A~ = −
~
.
5. Find the critical points for the systems in problems #1, 2, and 4. Then, we will look at how other solutions to these systems behave with respect to the critical points using a computergenerated phase plane.

BSU Math 333 (Ultman) Worksheet: Introduction to Systems of First Order Linear Differential Equations 5
PPLANE is a java-based app you can use to plot phase plane diagrams and trajectories. It can be a very useful tool. However, be advised: there are inherent risks anytime you download an application from the internet. If you do not want to risk downloading this file, you may want to try to find another way of plotting phase planes and trajectories.
◦ Download the pplane.jar from: http://math.rice.edu/~dfield/dfpp.html
.
◦ When you launch PPLANE, several windows will open. Select “OK” on the copyright window.
◦ Enter the equations defining your system in the “PPLANE Equation Window”. This window can also be used to adjust the display window.
◦ Clicking directly on a point in the window “PPLANE Phase Plane” will result in a graph of the trajectory (projection of a solution curve) passing through that point. You can also enter a point using the drop-down menu “Solution”, then selecting “Keyboard Input of Initial Value”.
◦ Use the “Keyboard Input of Initial Value” option to enter the x and y values for your critical point (equilibrium solution) to locate it on the phase plane. You can then click on points to examine the behavior of trajectories. This will be useful during your homework for section 3.2.