Some examples created by DifEqu

Lorenz's equation is one of the most tipical examples considered in the theory of ordinary differential equations. The first picture is the usual way of displaying the solutions: a projection to the (x,y) plane. The second picture is a 3 dimensional graphics of the solution. The third picture shows the x component of the solution as a function of t.

Another usual example is the damped pendulum. You can draw up the phase space with the direction field and the solution (first picture) and the energy function E=x*x+v*v (second picture). At the same time you can print out numerical values of t, x, v and E.

```           t              x              v             E
0.04000        9.99200       -0.39840       99.99879
0.08000        9.96813       -0.79299       99.99253
0.12000        9.92857       -1.18317       99.97632
0.16000        9.87349       -1.56835       99.94545
0.20000        9.80311       -1.94795       99.89540
0.24000        9.71766       -2.32143       99.82188
0.28000        9.61740       -2.68823       99.72088
0.32000        9.50260       -3.04781       99.58864
0.36000        9.37358       -3.39967       99.42172
0.40000        9.23063       -3.74331       99.21698
0.44000        9.07412       -4.07824       98.97160
0.48000        8.90438       -4.40399       98.68313
0.52000        8.72180       -4.72013       98.34943
0.56000        8.52677       -5.02622       97.96875
0.60000        8.31971       -5.32186       97.53969
0.64000        8.10103       -5.60665       97.06121
0.68000        7.87118       -5.88024       96.53266
0.72000        7.63061       -6.14227       95.95371
0.76000        7.37980       -6.39241       95.32444
0.80000        7.11922       -6.63038       94.64526
0.84000        6.84938       -6.85587       93.91692
0.88000        6.57076       -7.06864       93.14051
0.92000        6.28389       -7.26844       92.31745
0.96000        5.98928       -7.45506       91.44945
1.00000        5.68748       -7.62831       90.53854
1.04000        5.37902       -7.78801       89.58700
1.08000        5.06444       -7.93403       88.59737
1.12000        4.74430       -8.06623       87.57244
1.16000        4.41915       -8.18452       86.51518
1.20000        4.08954       -8.28881       85.42878
1.24000        3.75604       -8.37906       84.31658
1.28000        3.41922       -8.45524       83.18208
1.32000        3.07962       -8.51732       82.02889
1.36000        2.73783       -8.56534       80.86070
1.40000        2.39440       -8.59931       79.68128
1.44000        2.04988       -8.61930       78.49444
1.48000        1.70485       -8.62540       77.30401
1.52000        1.35985       -8.61769       76.11381
1.56000        1.01544       -8.59631       74.92762
1.60000        0.67215       -8.56139       73.74919
1.64000        0.33052       -8.51310       72.58216
1.68000       -0.00890       -8.45163       71.43009
1.72000       -0.34561       -8.37717       70.29641
1.76000       -0.67908       -8.28995       69.18441
1.80000       -1.00881       -8.19021       68.09721
1.84000       -1.33430       -8.07821       67.03776
1.88000       -1.65507       -7.95422       66.00882
1.92000       -1.97064       -7.81854       65.01294
1.96000       -2.28055       -7.67147       64.05243
2.00000       -2.58436       -7.51335       63.12939
```

A functional differential equation can be solved and drawn just as easily. For example, the solution of the equation x'(t):=-a(t)*h(x(t))*x(t)-g(t,x(t-r(t))) (a, h, g and r are user defined functions) is shown on the next picture:

So far DifEqu has two kind of partial differential equations built in: PDE's on the line or PDE's on the plane. Mathematically it means that the unknown function depends on t and x or t, x and y, respectively. PDE's with more than two variables can also be programmed in DifEqu, I just did not have the time. Let us see the string with non-zero boundary conditions and the membrane with zero boundary conditions:

Of course, all of the above are just examples of the use of DifEqu, one can solve other equations, display it a different way or can write a program for solving an entirely different problem.

If this program seems to be suitable for your research, download this program to find it out.