This is one of the sub-galleries of the ODE Gallery. See the ODE documentation for general information about ODEs and ODE galleries in 3D-XplorMath-J.

The exhibits in the "ODE(2D) 2nd Order" Gallery are second-order ordinary differential equations in two dimensions. Such an equation can be thought of as a system of two equations

x''(t) = f_{1}(t,x,y,x',y')

y''(t) = f_{2}(t,x,y,x',y')

where *x(t)* and *y(t)* are real-valued functions
of time, *t*. To specify an initial condition for the ODE, initial
values must be provided for *t*, *x*, *y*,
*x'*, and *y'*. (For the autonomous case,
*t* does not actually appear on the right-hand sides of the above
equations, and no initial value is required for *t*.)

To input an initial condition using the mouse, you must click-and-drag with the middle mouse button, or use the left mouse button while holding down the ALT key (the Option key on Mac's). The point where you click the mouse is the starting point of the solution curve, and the distance for which you drag the mouse determines the initial velocity. You can also input an initial point using the input boxes in the Control Panel.