## ODE(2D) 1st 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(2D) 1st Order" Gallery are first-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)

y'(t) = f_{2}(t,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*.)

For an autonomous first-order ODE in two-dimensions, the
pair of functions *(f*_{1}(x,y),f_{2}(x,y))
defines a *vector field*. A solution curve *(x(t),y(t))*
of the ODE satisfies the condition that its tangent vector at
each point is the value of the vector field at that same point.
In 3D-XplorMath-J, this vector field is visualized as a "direction
field." That is, a short line segment is drawn at each point in
a grid of points. The vector at that point lies along the displayed
line segment (but the length of the vector is not indicated in
any way). When a solution curve is drawn, the lines in the
direction field are tangent to the curve. (The direction field can
be hidden using the "Show Direction Field" option in the "Action"
menu.)

In the non-autonomous case, the pair of functions
*(f*_{1}(t,x,y),f_{2}(t,x,y)) defines
a *time-varying* vector field. This field
is visualized as a direction field that varies with time,
but *only while a solution curve is being drawn*.
For an example of this, see the non-autonomous User
Exhibit in this gallery.

To input an initial condition for the ODE using the mouse, you must click
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.
You can also input an initial point using the input boxes in the Control Panel.