Direction fields

This module deals with general first order ODEs, that is, equations of the form

$$\frac {dy}{dx} = f(x,y).$$

The module plots the associated direction (or slope) field, and it computes and plots numerical solutions of the equation. Exact solutions can also be calculated and plotted, provided Matlab's Symbolic Toolbox is installed.

After selecting Direction fields from the Iode main menu, the Direction fields window will open up. It shows a plot of the direction field for the ODE, with the equation itself written across the top of the plot.

When you first enter the module, you will find Iode has already chosen a default ODE as well as reasonable option settings. Next we explain how to change these settings and plot solutions.

Controls

• Solution method: specifies the method to be used when computing solutions. Numerical methods include Euler, Runge-Kutta, and Other (for user-defined methods, as in Appendix C); the final method is Exact (available provided the Matlab Symbolic Toolbox is installed). See also Remark 1.1 below.
• Step size: enter your desired step size (usually a small positive number) for the numerical solution method.
• Plot color: pick a color, any color, for your next solution plot. Note: Yellow tends not to show up very well.
• Initial conditions and Plot solution: You can tell Iode to plot a solution of the differential equation in two ways. Either enter the coordinates of the initial point into the Initial conditions boxes and then click on the Plot solutions button, or else simply click on the plot itself at the initial condition point (i.e., where you want the plot to begin). Note: In either case, Iode will compute (and plot) the solution curve both forwards and backwards from the initial condition.
• Exact solution: click here to try to obtain a formula for the general solution of the equation. This feature only works if the Matlab Symbolic Toolbox is installed. Also, most equations do not have explicit solution formulas anyway!
• In-graph controls:
  • By default, left-clicking in the graph will plot a solution through the click point. To change this default, see the Options menu below.
  • Dragging the mouse (i.e., moving the mouse while holding down the left button) over the graph creates a rectangle, and when you release the mouse button, Iode will zoom in on this rectangle.
  • Right-clicking in the graph will display a pop-up menu allowing you to erase solutions and undo zooms.

Remark 1.1   Plots of exact solutions can yield unexpected results. For instance, the exact symbolic solution of the default equation $\frac{dy}{dx}=sin(y-x)$ involves the inverse tangent atan. The atan function is only defined up to an additive multiple of $\pi$, and so the symbolic solution is only correct when the proper multiple of $\pi$ is added. Moreover, different multiples of $\pi$ might need to be added in different regions of the solution graph.

Now, when Matlab evaluates atan numerically, it always yields values between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$, which is usually the wrong choice and sometimes even results in discontinuous plots. Even worse, the plot of the numerical evaluation of a symbolic solution with initial values $(x_0,y_0)$ does not always pass through the point $(x_0,y_0)$ (!).This behavior is not strictly a bug in Matlab, but it is certainly a flaw that makes symbolic solutions less useful in practice than you might have expected.

Equation menu

• Enter differential equation: prompts you to enter the right hand side function $f(x,y)$ of the differential equation, using valid Matlab syntax -- consult Appendix C for examples. For example, to study the equation $dy/dx = yx^2$, you should input y*x^2. Notice you do not input the letter f, here. You just input the expression that you want on the right hand side of the ODE.
  Warning 1.2   Your expression for $f$ should involve the independent or dependent variables (or both), but no other variables!

• Change display parameters: prompts you to enter the domain and range of the plot, and the number of line segments to be shown in the direction field.
• Plot arbitrary function: allows you to plot any function for which you input the formula. For example, to plot $y=\sin x$ you just input sin(x). Consult Appendix C for more examples of functions you can use.

  equation, other

• Relabel variables: this prompts you for letters for the independent variable and dependent variable.

  When you input the names for the independent (horizontal) and dependent (vertical) variables, you are specifying the form in which you want the equation written: for example, $\bullet$

  • $dy/dx = f(x,y)$, in which case the solution will be a function $y$ of a variable $x$, or
  • $dx/dt = f(t,x)$, in which case the solution will be a function $x$ of a variable $t$.
  In the first case you would input x for the independent variable and y for the dependent variable. In the second case, input t for the independent variable and x for the dependent variable.

  Variable names should consist of only one letter, to avoid the risk of conflicting with Iode's internal variables.

Options menu

• Clicking on figure...: lets you choose what happens when you left-click on the plot. The four possibilities are
  • plots solution through click point (this is the default action),
  • zooms in on click point,
  • zooms out on click point,
  • recenters figure on click point.
  Incidentally, the time taken by the ``zoom out'' operation grows exponentially with the number of clicks because zooming out requires a recomputation of existing solutions over the new (larger) domain. If the domain doubles in size when you zoom out, then recomputing the solutions will take twice as long as previously because the number of steps has to be doubled also. If you zoom out again, the time needed to recompute all solutions doubles again, and so on.
• Change zoom factors: this only matters if you have set the clicking option to ``zoom''.
• Clear plot: to erase all curves.
• Refresh plot: try this if Matlab didn't update your plot properly. This sometimes happens when other windows partially cover a Matlab window.