Second order linear ODEs

This module deals with second order linear nonhomogeneous ODEs,

$$mx''(t) + c x'(t) + k x(t) = f(t) .$$ (3.1)

Equations of this sort arise in models of simple mechanical vibrations, and electrical circuits. (The application to mechanical vibrations explains why the module has filename mvgui.m.)

The module can plot the ``forcing'' function $f$, and it computes and plots numerical solutions of (3.1). The user can input the equation and the initial conditions, and choose the method used for computing solutions.

After selecting Second order linear ODEs from the Iode main menu, the Second order linear ODEs window opens up. The current ODE is displayed across the top, along with the current options for plotting solutions.

The options in this module are either self-explanatory or else are very similar to the corresponding parts of the direction fields module.

Controls

These are almost the same as in the Direction fields module (see Chapter 1), except for the default action of mouse clicks on the graph, which we now explain.

The initial conditions for solving a second order linear ODE consist of the initial time $t_0$, the initial position $x(t_0)$, and the initial velocity $x^\prime(t_0)$. You can give Iode this initial data in three ways.

First, you can enter the values $t_0, x(t_0)$ and $x^\prime(t_0)$ into the Initial conditions boxes, and then click on the Plot solution button.

Second, you can enter the desired initial slope $x^\prime(t_0)$ into the relevant Initial conditions box, and then click on the graph where you want the solution to start. Iode will plot a solution through the click point whose initial slope is the value you entered in the box.

Third, you can press down the mouse button at the desired initial point in the graph and then drag the mouse a short distance at the desired slope. When you release the mouse button, Iode will plot a solution starting at the point where you first pressed the button, with the initial slope of that solution being given by the line going from where you first pressed the button to where you finally released it. Play around and see how it works!

Equation menu

When you select Enter differential equation you will be prompted for the coefficients $m,c$ and $k$ to be used in equation (3.1). These coefficients are allowed to involve the independent variable, but are often just constants. You will also be prompted to enter the forcing function $f$.

Otherwise this menu is very similar to the corresponding part of the direction fields module, described in Chapter 1.

Options menu

This menu is the same as the corresponding part of the direction fields module, described in Chapter 1, except for one additional menu item, Show forcing function, used for either displaying or hiding the plot of the forcing function.