## ODE(3D) 2nd Order Gallery

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(3D) 2nd Order" Gallery are second-order ordinary differential equations in three dimensions. Such an equation can be thought of as a system of three equations

x''(t) = f1(t,x,y,z,x',y',z')

y''(t) = f2(t,x,y,z,x',y',z')

z''(t) = f3(t,x,y,z,x',y',z')

where x(t), y(t), and z(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, z, x', y', and z'. (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. The pre-defined exhibits are autonomous ODEs, but a User Exhibit is provided for the non-autonomous case.)

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. If you simply click the mouse, without dragging it, then the initial velocity is zero. Note that mouse input is limited to the plane of the screen. The initial point is a point on that plane, and the initial tangent vector lies in that plane. You can input any initial point and tangent vector using the input boxes in the Control Panel.

A solution of the equation is a curve (x(t),y(t),z(t)) in three-space. Solution curves are shown as three-dimensional curves, which can be rotated and displayed in stereo view, just like any other 3D exhibit.