## ODE(1D) 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(1D) 2nd Order" Gallery are second-order ordinary differential equations in one dimension. All the exhibits in this gallery are autonomous ODEs; that is, the functions that define the ODEs do not depend on time, t. The equation that defines an autonomous second-order ODE in one dimension has the form

x''(t) = f(x,x')

where x(t) is a real-valued function. A solution x(t) has its values in the real numbers and could be visualized as a point that moves along a line as t changes. However, it is also possible to display a two-dimensional visualization in which the point (x(t),x'(t)) in the plane is plotted for each value of t, and this is the visualization that is used in 3D-XplorMath-J.

Note that by introducing the function y(t) = x'(t), we can write the above equations an equivalent system of two equations:

x'(t) = y(t)

y'(t) = f(x,y)

This system represents a first-order ODE in two dimensions, so that a second-order 1D ODE is equivalent to a first-order 2D ODE. The visualizations used for these two types of ODEs are the same in 3D-XplorMath-J. In particular, a direction field can be displayed as part of the visualization to show the direction of the tangent vector at each point along a solution curve. (See the documentation for the ODE(2D) 1st Order Gallery for more information.)

To specify an initial condition for the ODE, initial values must be provided for x and x'. You can input an initial point using the input boxes in the Control Panel. Alternatively, you can use the mouse to select an initial point by clicking with the middle mouse button, or by using the left mouse button while holding down the ALT key (the Option key on Mac's).