**DiffEqs `**DEGraphics

This package provides the following functions for plotting solutions of differential equations.

Plotting solutions of differential equations. **DEPlot**[*f*[*t, y*], {*t*, *tmin*, *tmax*}, {*y*, *ymin*, *ymax*}] creates a plot showing the direction field and solution curves for the differential equation *y*' = *f*[*t, y*]. **PhasePlot**[{*f*[*x, y*], *g*[*x, y*]}, {*t*, *tmin*, *tmax*}, {*x*, *xmin*, *xmax*}, {*y*, *ymin*, *ymax*}] creates a plot showing the direction field and solution curves for the autonomous system *x'* = *f*[*x, y*], *y*' = *g*[*x, y*]. **PhasePlot**[{*f*[*t, x, y*], *g*[*t, x, y*]}, {*t*, *tmin*, *tmax*}, {*x*, *xmin*, *xmax*}, {*y*, *ymin*, *ymax*}] creates a plot showing solution curves for the system *x'* = *f*[*t, x, y*], *y*' = *g*[*t, x, y*]. **NDPlot**[*eqns*, *fns*, {*t*, *tmin*, *tmax*}] uses NDSolve to compute a numerical solution of a system of (up to 3) first-order differential equations and plots the solution. Returns {*solution*, -Graphics-}. **PoincareTimeSection**[{*f*[*t, x, y*], *g*[*t, x, y*]}, {*t*, *t*_{0}, *tmax, dt*}, {*x*, *x*_{0} }, {*y*, *y*_{0 }}] creates a Poincaré time section plot of the solution of *x'* = *f*[*t, x, y*], *x*[*t*_{0}] = *x*_{0}, *y*' = *g*[*t, x, y*], *y*[*t*_{0}] = *y*_{0}. **ViewProjections**[{*f*[*t*], *g*[*t*], *h*[*t*]}, {*t*, *t*_{0}, *tmax*}, {*x, y, z*}] creates a GraphicsArray of projections of the curve (*f*[*t*], *g*[*t*], *h*[*t*]) onto the *xy*, *xz*, and *yz* coordinate planes. **PlotImplicit**[*F*[*t, y*], {*t*, *tmin*, *tmax*}, {*y*, *ymin*, *ymax*}] creates a contour plot of the function *F*. **TimeStatePlot**[{*x*[*t*], *y*[*t*]}, {*t*, *tmin*, *tmax*}, {*y*, *ymin*, *ymax*}] creates a 3D plot of the ``time-state trajectory" (*t*, *x*[*t*], *y*[*t*]) with projections onto the *t x*, *t y*, and *x y* planes.

This loads the package.

`<<DiffEqs`DEGraphics``

**DEPlot **

Here are a few examples of the use of DEPlot. Each of these uses the differential equation .

DEPlot accepts any of the options of ListPlot, as well as the following:

Curve options for `DEPlot`.

With InitialPoints -> Automatic, DEPlot uses four uniformly spaced initial points to draw four curves. With InitialPoints -> None or MaxCurves -> 0, no curves are drawn. If MaxCurves -> *n*, where *n* > 0, then, for each initial point, DEPlot attempts to generate additional starting points if *n* is odd and *n* additional starting points if *n* is even. This is done by taking steps orthogonal to the direction field.

Arrow style options for `DEPlot`.

Display options for `DEPlot`.

Numerical options for `DEPlot`.

Standard (nonadaptive) fourth-order Runge-Kutta is the default numerical method. Heun is the second-order method that is often called the improved, or modified, Euler method. With Method -> DormandPrince, the eighth-order (nonadaptive) Dormand-Prince method is used. Method -> Automatic causes DEPlot to use NDSolve successively over intervals [*t*, *t*+*h*], where *h* = StepSize. Thus StepSize does not affect the accuracy in this case--only the smoothness of the curve drawn. Method -> Automatic is generally much slower at a given stepsize, but for difficult (i.e., stiff) equations, may give much better results. When Method -> Automatic, any of the options of NDSolve can be used.

This illustrates the ShowEquilibriumPoints option.

Converted by