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) 1st Order" Gallery are first-order ordinary differential equations in one dimension. The equation that defines such an ODE has the form

x'(t) = f(t,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, in 3D-XplorMath-J, the
solution is visualized by displaying the *graph* of the solution
function *y = x(t)*. That is, the point *(t,x(t))* in
the plane is plotted for each value of *t*. Note that in
this gallery, the value of the solution function *x(t)* at time *t*
is represented by the *vertical* axis, while the horizontal axis
represents time. A direction field
is also shown by default as part of the visualization. This field
represents the slope of the tangent line to the solution
curve at each point. (The direction field can be hidden using a command in
the "Action" menu.)

To specify an initial condition for the ODE, initial
values must be provided for *t* 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).