Using Mathcad to Solve Systems of Differential Equations
Charles Nippert
Getting Started
Systems of differential equations are quite common in dynamic simulations. Solving a
system of differential equations is somewhat different than solving a single ordinary
differential equation. The solution procedure requires a little bit of advance planning.
The system of differential equations must first be placed into the "standard form" shown
below:
y'1 = F1 (t , y1 , y 2 L y n )
y' 2 = F2 (t , y1 , y 2 L y n )
M
y' n = Fn (t , y 0 , y1 L y n )
y 1 (t 0 ) = C 1
y 2 (t 0 ) = C 2
M
y n (t 0 ) = C n
It is generally possible to do this transformation however it may require some
manipulation.
Example of Transformation of a System of Differential Equations
Consider the system of differential equations and initial value conditions shown below
y' '1 = y1 − 3y 2
y 2 = y1 '+ y' 2
y1 (0) = 2
y'1 (0 ) = 1
y 2 (0 ) = 3
To generate a solution, define a new variable y3 = y1’ and rearrange to obtain the
following system.’
y'1 = y 3
y' 2 = y 2 − y 3
y' 3 = y 1 − 3y 2
y1 (0) = 2
y 2 (0 ) = 3
y 3 (0) = 1
In the sample on these notes you will obtain a numerical solution for the following
system of differential equations. You will the use the solution to make both a
conventional parametric plot and a plot of the phase plane.
y'1 = y 2
y' 2 = y1
y1 (0) = 1
y 2 (0) = 2
Enter the Initial Values
1.
Open Mathcad in the usual fashion and move the cursor about 1 inch below the
menu slightly to the right. If the matrix toolbar is not visible open it by choosing
"View/Toolbars/Matrix". The initial values are entered as a vector. You can give
the vector any name you wish. In this example enter the name space "y0". Press
the ":" key to generate the arithmetic assignment symbol ":=". Use the mouse
cursor to press the matrix logo on the matrix toolbar,
. On my toolbar it is the
first button on the first row. The "Insert Matrix" dialog box immediately appears.
The initial conditions are entered as a vector with one column and two rows.
When you are finished the dialog box should look like figure 1.
Figure 1
Insert Matrix Dialog Box
2.
Press the "OK" button. The dialog box should immediately close and a column
vector with two elements should immediately appear. The blue inverted L. cursor
should be on the upper black rectangle or placeholder in the matrix. Type "1" to
enter the first boundary condition. You can use the arrow keys on the keyboard to
move between the elements on a matrix. Press the down arrow key to move to the
next element and enter its value, "2". The finished vector should look like figure
2.
Figure 2
Initial Condition Vector
3.
Press "Enter". The system of differential equations is entered as a vector
function. Because this vector function is somewhat large, you may wish to move
the cursor down to allow enough space for it. Mathcad uses a standard function
format similar to the kind in used in many programming languages. First the
function name is entered followed by parentheses that enclose the "argument list"
which is a list of variables the function needs to perform its operations. The
argument list for the vector function containing the derivatives consists of the
independent variable (the parameter in the dominator of the derivative), and a
vector that will contain the values of the dependent variables. You may give the
vector function any name you wish. In this example name the vector function
"D". Enter the left-hand side of the vector function definition by typing "D(t, y):".
Create a vector with one column and two rows by pressing the matrix button. The
number of rows and columns in the dialog box should be these values because the
dialog box should retain the values you entered when you entered the boundary
conditions in the initial conditions vector. Your screen should now look like
figure 3.
Figure 3
Entering The Differential Equations
4.
Recall that the differential equations are
y' 0 = y1
y'1 = y 0
Now, enter the right hand side of the first differential equation, the one for y’0.
First press "y", this is the name of the vector that you entered in the argument list.
To enter the subscript, press the subscript key,
. In my computer this button
was to the right of the matrix button, and was the second button on the first row.
The small black rectangle will appear below in slightly to the right of the letter y.
Press the "0" key data value the subscript. Use the down arrow key to move to
the second placeholder and enter the second differential equation. Follow the
procedure you followed in the first differential equation to enter the subscript.
When you have completed entering the differential equation, it should look
something like figure 4.
Figure 4
The Completed System Of Differential Equations
5.
Mathcad has several functions for solving systems of ordinary differential
equations. Each one uses a different integration algorithm and takes the same
arguments. The three functions are shown in the table below:
Table 1
Mathcad Functions For Solving Systems Of Ordinary Differential Equations
Function Name
Brief Description
Rkfixed
Runge-Kutta method with a fixed step
Rkadapt
Runge-Kutta method with a variable step (a more robust
method that adjusts the integral step size based on an
estimation of integration error)
Bulstoer
Bulirsh-Stoer method (a very robust method which some
prefer to Runge-Kutta)
All three functions take the same argument list which is shown below
Table 2
Sample Reference To Mathcad’s Differential Equation Solvers
rkfixed(y0,ti,tf,M,D)
Rkadap(y0,ti,tf,M,D)
Bulstoer(y0,ti,tf,M,D)
Variable
Description
y0
The name of the vector containing the initial conditions
ti
The starting point of the integration
tf
The ending point of the integration
M
The number of integration steps between the starting and
ending point
D
The name of the vector function containing the differential
equations
The differential equation solvers are called on the right hand side of an arithmetic
assignment statement. The left-hand side defines a matrix that will contain the
results of the numerical integration. In this example, you will create a matrix
named "yn", use the "rkfixed" function to obtain a solution from zero to two using
20 integration steps. Press "Enter" if your screen still shows the blue inverted L.
cursor in the differential equation vector. The equation you're going to enter will
APPEAR on the screen as "yn := rkfixed(y0,0,2.0,20,D)". Remember to type ":"
or use the evaluation toolbar to make the arithmetic's assignment symbol, ":=",
appear on screen. Your completed solution should look like figure 5.
Figure 5
Completed Mathcad Solution to a System of Ordinary Differential Equations
6.
You can see the numerical results of the integration by displaying the contents of
the matrix "yn". Press enter and move the red cross cursor far down and to the
right of the rest of your equation. Next, type "yn=". A table showing a portion of
the results will appear. Clicking on any portion of the table will cause that portion
to be highlighted and will cause a scrollbar to appear to the right of the table. Use
this scrollbar to make the rest of the table visible. This table is shown in figure 6.
Figure 6
Table Showing Partial Results Of Numerical Integration
Clicking the mouse
cursor on any
portion of the table
highlights that
portion and makes
the scroll bar
visible.
Use the scroll
bar to view
any portion of
the table.
8.
Notice that the table in figure 6 has three columns. The first column shows the
range of the independent variable. The next column shows the values of y0. The
final column shows the values of y1. Often this tabular data is all that you need.
However it is often desirable to display the data in graphical form. It is important
that you notice the shift in subscripts that Mathcad makes. As you enter the
integration the first dependent in variable as the subscript zero. However, the first
dependent variable has the subscript one in the output matrix. Before you make
the plot, you will make 3 1-dimensional arrays that contain a date in the columns
of the yn matrix. Doing this step will make your plots easier to read and create.
Entering individual columns can prove difficult if you intend to draw more than
one line on a plot. Select an area of your worksheet somewhere below the
differential equations and click the mouse button. Type "t:yn" Type "yn" then
on the matrix toolbar. This button was the
click on the column range button
second button in the second row of the matrix toolbar in my computer. Type "0"
to specify the column number in the yn matrix. At this point the arithmetic
assignment statement should look like figure 7.
Figure 7
Defining a Vector From a Column In a Matrix
8.
Create two more arrays, y1 and y2 that contain the values from the first and
second columns of the yn matrix. This portion of your worksheet should now
look like figure eight.
Figure 8
Defining the Vectors From the yn Matrix
t := yn<0>
y1 := yn<1>
y2 := yn<2>
9.
You will now make a plot of y0 and y1 versus the independent variable, t. Move
the red cross cursor to a vacant area of your worksheet. You can create a x-y plot
in any of the following ways:
1.
Press the "@"
2.
Select "Insert/Graph/X-Y Plot" from the menu.
This plot will be y1 and y2 versus t. Enter "t" as the name of the variable on the
horizontal axis. Keep pressing the left arrow key until the blue inverted L. cursor
surrounds the leftmost solid black rectangle that is the placeholder for the vertical
axis. When plotting more than one line on a X-Y axis, you enter the names of the
arrays separated by a coma. Type the characters "y1,y2" and press "Enter". The
graph should immediately appear. Double-click on the graph to call up the
formatting dialog box. Your worksheet should now resemble figure 9.
Figure 9
The Solution To This System of Differential Equations
10.
You can now format this graph. First, check the "Grade to Lines" box for both
the x-axis and y-axis. The analytical solution of these differential equations
includes exponential terms check the "Log Scale" option for the y-axis. Press
"OK". The dialog box should immediately close and the graph will show grid
lines and the y-axis will become a logarithmic axis. Use the mouse to grab the
solid black rectangular "handle" in the lower right corner and enlarge and the
graph. Now your figure should resemble figure 10.
Figure 10
Finished Plots of y1 & y2 versus t
11.
A phase plot is a plot of the dependent variables plotted against each other. The
independent variable is not shown. You will not make a phase plot for this
system. Click on a portion of the worksheet that is below your solution but still
empty and create a new graph. This time label the horizontal axis "y1" and label
the vertical axis "y2". Double-click on the graph to open the format dialog box
and check "Grid Lines" for both axes. Press the "OK" button and then enlarge the
graph. Your face plot should resemble figure 11.
Figure 11
Finished Phase Plot