SYSTEMS OF DIFFERENTIAL EQUATIONS

Graphmatica can also approximate the solutions of up to fourth-order systems of ordinary differential equations. [Refer to your local textbook for more background information.] You can pose your system in either of two sets of variables: { t, x, dx, y, dy, z, dz, w, dw} or { t, x1, dx1, x2, dx2, x3, dx3, x4, dx4 }. In each case, t is the independent variable, and the variables dX are the derivatives dX/dt. You specify the complete system in a manner similar to parametric equations, namely (for a fourth-order system):

dx=f1(x,y,z,w,t); dy=f2(..); dz=f3(..); dw=f4(..) {t0,x0,y0,z0,w0}

where f1...f4 are some functions of the variables x,y,z,w, and t, and x0...w0 are the initial values of the system at t0. (You need not isolate the differentials on the left of the equals sign, but you can only use one per equation.) Separate equations with semi-colons, and enclose the initial values in curly braces.

If you want to graph a third- or second-order system, just leave out the equations for dw, or for dz and dw, and use the appropriate number of initial values instead.

Note that to simplify the process of parsing the system and associating each equation in the system with a graph, the order of equations dx, dy,dz,dw (or dx1, dx2, dx3, dx4) is required, and the program will not graph a system whose equations are listed in the wrong order.

The values for each curve in the system are calculated simultaneously, but only the curve for x(t) is drawn as it is calculated; the others are displayed as soon as calculation is finished. Each curve plots in a different color, which corresponds to the color of its equation in the system when it is listed in the status bar.

You can also choose the initial value points using the mouse. See Setting the initial value... for details.