Direction Fields in Mathematica

[steelphantom]

I just picked up a copy of Mathematica through Penn State, and I'm trying to figure out how to plot a direction field of a differential equation. For example, I have the differential equation $$dv/dt = 32 - 8v$$

I've found http://support.wolfram.com/mathematica/graphics/2d/directionfield.html" on Wolfram's site that shows you how to do it, but in the line

In[3]:= field=PlotVectorField[{1,Last[eqn]},{x,-2,2},{y[x],-2,2}]

kind of confuses me with the arguments that are used. I understand the second two sets of arguments (x and y arguments), but what's up with the first one {1, Last[eqn]}? Where did the 1 and Last come from?

Also, there's another page on Wolfram's site that displays the following as the format for the PlotVectorField function:

PlotVectorField[f, {x, x0, x1, (xu)}, {y, y0, y1, (yu)}, (options)]

What do (xu) and (yu) represent? The rest of it I understand (I think! ). Sorry for all of these questions. I'm definitely a Mathematica n00b and I think it's going to take a little getting used to. Thanks!

 

[Crosson]

First of all I suggest you always use the built in Mathematica documentation until you know lots and lots of what's in there.

The following code will do what you want:

(* Loads the Package *)

<< Graphics`PlotField`

(* The one is the t_component of the vector field and Last[
  Eqn] gives the right_hand_side of and equation i.e. the v_component. I put \
this in directly *)


Field = PlotVectorField[{1, 32 - 8v}, {t, 0, 10}, {v, -5, 5}]

(* Here is some more code that will impose a solutions on the vector field assuming that you also ran the code above*)

Show[Field, Plot[Evaluate[v[t] /. NDSolve[{v'[t] == 32 - 8v[t], v[0] == -6}, \
v, {t, 0, 20}]], {t, 0,
   10}, PlotStyle -> Red, PlotRange -> {{0, 10}, {-5, 5}}]]

 

[tacman]

Hello there,

I can definitely understand your confusion with the arguments used in the PlotVectorField function. Let me try to break it down for you.

Firstly, the {1, Last[eqn]} argument represents the vector field function. In this case, it is simply the vector <1, 32-8v>. The number 1 is used to represent the x-component of the vector, while Last[eqn] represents the y-component. In this case, eqn is the differential equation dv/dt = 32 - 8v, and Last[eqn] simply extracts the last part of the equation, which is -8v.

The (xu) and (yu) arguments represent the grid spacing for the x and y axes, respectively. This allows you to control the density of the vectors in the direction field. The larger the value, the more sparse the vectors will be, while a smaller value will result in a denser field.

I hope this helps to clarify things for you. Mathematica can definitely be overwhelming for beginners, but with practice and some guidance, you'll soon get the hang of it. Good luck!