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) = f1(t,x,y)
y'(t) = f2(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 (f1(x,y),f2(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 (f1(t,x,y),f2(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.