Numerical Solutions to Systems of Differential Equations

You can solve the following system numerically by entering the equations into a 6×1 matrix and choosing Numeric from the Solve ODE submenu. Three functions x, y, and z are returned as output.


$\blacktriangleright$ Solve ODE + Numeric

x = x + y - z
y = - x + y + z
z = - x - y + z
x(0) = 1
y(0) = 1
z(0) = 1
, Functions defined: x, y, z


The following table lists values of x, y, and z as the independent variable t varies from 0 to 1.

t x y z
0 1.0000 1.0000 1.0000
.1 1.1158 1.0938 .8842
.2 1.2668 1.1695 .7332
.3 1.4582 1.2173 .5418
.4 1.6953 1.2253 .3047
.5 1.9830 1.1791 .0170
.6 2.3256 1.0619 - .3256
.7 2.7265 .8542 - .7265
.8 3.1873 .5344 -1.1873
.9 3.7077 .0777 -1.7077
1.0 4.2842 - .5424 -2.2842


You can create a matrix with these values as follows.

1.
Apply Evaluate to x$\left[\vphantom{
\begin{array}{r}
0.0 \\
.1 \\
.2 \\
.3 \\
.4 \\
.5 \\
.6 \\
.7 \\
.8 \\
.9 \\
1.0
\end{array}
}\right.$$\begin{array}{r}
0.0 \\
.1 \\
.2 \\
.3 \\
.4 \\
.5 \\
.6 \\
.7 \\
.8 \\
.9 \\
1.0
\end{array}$$\left.\vphantom{
\begin{array}{r}
0.0 \\
.1 \\
.2 \\
.3 \\
.4 \\
.5 \\
.6 \\
.7 \\
.8 \\
.9 \\
1.0
\end{array}
}\right]$ to get x$\left[\vphantom{
\begin{array}{r}
0.0 \\
.1 \\
.2 \\
.3 \\
.4 \\
.5 \\
.6 \\
.7 \\
.8 \\
.9 \\
1.0
\end{array}
}\right.$$\begin{array}{r}
0.0 \\
.1 \\
.2 \\
.3 \\
.4 \\
.5 \\
.6 \\
.7 \\
.8 \\
.9 \\
1.0
\end{array}$$\left.\vphantom{
\begin{array}{r}
0.0 \\
.1 \\
.2 \\
.3 \\
.4 \\
.5 \\
.6 \\
.7 \\
.8 \\
.9 \\
1.0
\end{array}
}\right]$ = $\left[\vphantom{
\begin{array}{l}
1.0 \\
1.1158 \\
1.2668 \\
1.4582...
...3256 \\
2.7265 \\
3.1873 \\
3.7077 \\
4.2843
\end{array}
}\right.$$\begin{array}{l}
1.0 \\
1.1158 \\
1.2668 \\
1.4582 \\
1.6953 \\  
...
...3 \\
2.3256 \\
2.7265 \\
3.1873 \\
3.7077 \\
4.2843
\end{array}$$\left.\vphantom{
\begin{array}{l}
1.0 \\
1.1158 \\
1.2668 \\
1.4582...
...3256 \\
2.7265 \\
3.1873 \\
3.7077 \\
4.2843
\end{array}
}\right]$

2.
Similarly, apply Evaluate to get the y and z columns.

3.
To create a matrix with all four columns, place the t, x, y, and z columns next to one another, and from the Matrices submenu choose Concatenate.

4.
To add a row at the top for labels, select the matrix and choose Edit + Insert Rows. (You must select only the matrix, not including brackets, to have Insert Rows appear on the Edit menu.)

1.
To line up entries, select a column, choose Edit + Properties, and change Column Alignment to Aligned Left or Aligned Right.

You can also take advantage of the fact that you are using a word processor to put the values into a 12 by 4 table. You can then revise the table to add lines.