Given either a system of two first order autonomous differential equations, or a single first order differential equation, dfieldplot produces a direction field plot. There can be only one independent variable.

For plotting solution curves, see DEplot or phaseportrait.

The direction field presented consists of a grid of arrows tangential to solution curves. For each grid point, the arrow centered at ($x,y$) will have slope $\frac{\mathrm{dy}}{\mathrm{dx}}$. For system of two first order autonomous differential equations this slope is computed using $\left(\frac{\mathrm{dy}}{\mathrm{dt}}\right)$/$\left(\frac{\mathrm{dx}}{\mathrm{dt}}\right)$, where these two derivatives are specified in the first argument to dfieldplot. Note: A system is determined to be autonomous when all terms and factors, other than the differential, are free of the independent variable. For more information, see DEtools[autonomous]. For the single first order differential equation, the slope is simply $\frac{\mathrm{dy}}{\mathrm{dx}}$.

Optional equations for dfieldplot are in the DEplot help page.

The xrange and yrange parameters must be specified as follows.

By default, integration along a solution curve stops one mesh point after the specified range is exceeded.  This may be overridden by the obsrange option.

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$\mathrm{dfieldplot}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)=\frac{1}{2}\left(-x-{\left(x^2+4y\left(x\right)\right)}^{\frac{1}{2}}\right)\,y\left(x\right)\,x=-3..3\,y=-3..2\,\mathrm{title}=\mathrm{`Restricted\; domain`}\,\mathrm{color}=\frac{1}{2}\left(-x-\left(x^2+4y\right)\right)\right)$

$\mathrm{dfieldplot}\left(\left[\mathrm{diff}\left(x\left(t\right)\,t\right)=x\left(t\right)\left(1-y\left(t\right)\right)\,\mathrm{diff}\left(y\left(t\right)\,t\right)=0.3y\left(t\right)\left(x\left(t\right)-1\right)\right]\,\left[x\left(t\right)\,y\left(t\right)\right]\,t=-2..2\,x=-1..2\,y=-1..2\,\mathrm{arrows}=\mathrm{SLIM}\,\mathrm{color}=\mathrm{black}\,\mathrm{dirfield}=\left[10\,10\right]\right)$