### Systems of Differential Equations

Graphs solution functions x(t) and y(t) to the system of differential equations x '=f(x,y) and y '=g(x,y), with initial values given by x(t0)=x0 and y(t0)=y0.
How to use   ||   Examples   ||   Other Notes

How to use
• Enter the function f (x,y) for x' in the text input field marked "x'=f (x,y)=" (Example: x'=y)
• Enter the function g(x,y) for y' in the text input field marked "y'=g(x,y)=" (Example: y'=x)
• Click the "Graph" button (this button also refreshes the graph)
• The initial values x0, y0, and t0 can be changed in either of two ways:
• Enter the values directly into the text input fields marked "x0=", "y0=", and "t0="
• Click and drag the red points on the graph -- horizontally, the red points move together, and their horizontal position corresponds to the value of t0; vertically, the points move independently, with the vertical position of the point on the blue graph corresponding to the value of x0, and the vertical position of the point on the green graph corresponding to y0
(Example: x0=0, y0=1, and t0=0)
• To erase the graph and the text input fields for f (x,y) and g(x,y), and set x0, y0, and t0 to default values, click the "Clear" button

The text input fields for f (x,y) and g(x,y) can accept a wide variety of expressions to represent functions, and the buttons under the graph allow various manipulations of the graph coordinates. The text input fields for x0, y0 and t0 can accept real numbers in decimal notation.

Examples
 Dual Oscillation:     x'=y y'= -x x0=0, y0=1, t0=0 Predator-Prey Model: x'=x(1-y) y'= y(x-1) x0=3, y0=1, t0=0

Other Notes
The graph shows solution functions x(t) andy(t) to the system of differential equations x'=f (x,y) and y'=g(x,y), with initial values given by x(t0)=x0 and y(t0)=y0. The particular solution functions x(t) and y(t) to the system of differential equations satisfying the given initial values will be graphed in blue (for x(t)) and green (for y(t)). Since the functions f (x,y) and g(x,y) do not depend on the variable t, changes in the initial value t0 only have the effect of horizontally shifting the graphs.

The particular solution functions x(t) and y(t) are graphed by determining numerical approximations to the functions using the classical (order four) Runge-Kutta Method.

In the "Predator-Prey Model" example above the particular solution functions represent populations in a simple two-species predator-prey model. The function x(t) (in blue) represents the prey population, and y(t) (in green) represents the predator population -- with the given initial values x0=3 and y0=1, note the periodic population explode-collapse pattern, with the predator population lagging the prey population (with prey population high and predator population low, prey population can grow, but then plentiful food supply makes predator population also grow; overpredation causes prey population to collapse, resulting in starvation and collapse for the predator population; low predator population allows the prey population to recover, and the cycle repeats). The two populations do have equilibrium values, associated with x0=1 and y0=1 at time t0. It is also instructive to look at the phase portrait for this system.