Second-Order Ordinary Differential Equation(ODE)---Generic Form (version 1.4)

This page/program provides numerical solution of a 2nd-order ODE in the form of 2 equations:
          dy0/dt=f0(y0,y1,t),
          dy1/dt=f1(y0,y1,t),
          at t=t0, y0(t)=y0(t0), y1(t)=y1(t0).

In math equation style:
`dy_0/dt=f_0(y_0,y_1,t),`
`dy_1/dt=f_1(y_0,y_1,t),`
at `t=t_0, y_0(t)=y_0(t_0), y_1(t)=y_1(t_0)`

Another form of 2nd-order ODE, `y''=f(y, y', t)`, is a special case of the above 2-equation form. 
Let `y=y_0, y'=y_1,` then `f_0(y_0,y_1,t)=y_1,` and `f_1(y_0,y_1,t)=f(y,y',t)=f(y_0,y_1,t)`

When you open this page, the solution/simulation of a default/pre-defined example ODE is ready to run.
Click "Solve" button to display the solution for the ODE over a time period: [t0, t0+T], where T=N*dT.
A set of ODE examples are provided, use the selection menu to select example functions, and then click "Solve" button to run the new example.
Click on the "Phase Plane" equal to: (1)-Set the respective initial values as the clicked point position. and (2)-Same as click "Solve" button to start a new ODE solution process with the new initial values.
A list of examples is at the end of this page.

To define your own ODE system, just input or re-define the ODE functions and parameters:
Functions: f0(y0,y1,t), and f1(y0,y1,t).
Initial values: y0(t0),and y1(t0).
dT--Integration/sampling step-size, N--Number of data to display in the 2 plotters. Then, click "Solve" button to start your own ODE solution/simulation.
A more detailed usage of this ODE tool is at the end of this page.
For definition of the functions f0(y0,y1,t), and f1(y0,y1,t), see "Supported Operators and Functions".

Click "Solve" Button to Run the Predefined Example

Example list of the ODE Tool:

This program provide a set of ODE examples to demo the function f0(y0,y1,t) and f1(y0,y1,t)
input syntax. Use the "Menu: Select,,," to see the examples and then click "Solve" button to run the selected example.
Example default: a van der Pol oscillator with time-varying parameters.
Example 1: a non-linear, time-varying system.
Example 2: a van der Pol oscillator with time-varying oscillating frequency.
Example 3: a harmonic oscillator, free, no damping, with a time-varying parameter.
Example 4: a time-varying system, driven by sine time functions.
Example 5: a harmonic oscillator, with damping, and time-varying parameters.

More Usage Information for the ODE Tool:

This program provides numerical ODE solution over a definite time period [t0, t0+T], where T=N*dT.
If f0(y0,y1,t) or f1(y0,y1,t) function depends only on t, the corresponding output y0(t) or y1(t) will become an integral of the function f0(t) or f1(t). 
If f0(y0,y1,t) and f1(y0,y1,t) functions depend on y0, y1, not on t, the ODE will become autonomous(or time-invariant). 
The classic explicit fourth-order Runge-Kutta method(RK4) is used as numerical integration method in this program.

Usually the value of N can be set between 1000 to 20000. A bigger N will display more data in the graph plotter, but will also cost more memory or even cause a stack overflow problem, and hence stop the program.
Currently, the ODE solver/tool employs a 'fixed step size' integration method and the step size dT must be set by the user. Theoretically, a proper value of step size dT can be determined according to the ODE's maximum state changing rate, maximum inherent frequency(or minimum period), minimum time-constant, and/or the so-called stiffness.
As a practical approach, you can begin with a small dT and use the "Solve" button, to get initial yi(t)(i=0,1) displays, and then use the yi(t) displays to obtain a rough estimate of the ODE's minimum time-constant(or minimum period).
Usually dT can be set between 1/20 to 1/5 of the minimum time-constant (or minimum period) for most non-stiff ODEs. A smaller dT will lead to higher accuracy, but will cause longer computation time as well.
You can try different combinations of dT and N values, to get a stable and accurate ODE solution, and also nice graph displays, at the same time.
A special tool for exploring relationship between the step size and accuracy of the ODE solution will be developed and added to our website.
In the future, the so-called 'adaptive step size' algorithms may be added to our ODE solvers/tools. Then, the step size will be variable during the integration process according to the dynamical characteristics of the ODE. That is, the step size dT will be automatically determined by the ODE solver itself, and the user just need to set error tolerance parameters optionally.

After phase plane coordinates are established, move the Mouse cursor on the phase plane will display the [y0, y1] position. Click the Mouse left key, will start a "Solve" process with the displayed [y0, y1] as the initial condition of the ODE solution, that is the solution trajectory will start from the [y0, y1] point. Notice that both the phase plane and oscilloscope plotters are auto-scaled and auto-centered, the clicked initial point [y0, y1] point may be displayed in a new coordinate grid immediately. Before click again, you need to move the Mouse to update the mouse position on the phase plane.
The phase plane graph is auto-scaled and auto-centered on the canvas. If the checkbox "Keep Graph Shape or Not" is checked, the plotter uses the same scale factor in both x and y directions and the graph will keep its original shape. If the checkbox is unchecked, the graph will be stretched in one direction to make full use of the canvas area.
On the phase plane graph, a series of colored dots, from blue to red, are used to indicate the moving direction and/or moving speed of the solution points on the plane.
The solutions y0(t) and y1(t) will be displayed in both the oscilloscope and phase plane plotters.
If you changed any input parameter, function, and/or checkbox, you need to click "Solve" again to re-start and let the new parameter, function selection and/or checkbox status be read and used in the new solution.
In case an error occurs, the program will display a notice and may indicate the cause of the error. After the error is corrected, Click "Solve" button to re-start a new solution process.