Example: y'=y
Click the "Graph" button (this button also refreshes the graph).
Choose the initial value point (x_{0},y_{0}).
Example: (x_{0},y_{0})=(0,1)
This can be done either of two ways:
Exponential Growth: y'=y
Exponential Decay: y'=-y
Logistics Growth: y'=y(1-y)
Semicircles: y'=-x/y
Radial Lines: y'=y/x
The particular solution function y(x) is graphed by determining a numerical approximation to the function using the classical (order four) Runge-Kutta Method (which, in the case where the function f (x,y) is actually a function of the single variable x, reduces to Simpson's Rule for integrating functions of the form f (x) -- see the function of x only example above).
Since this method encounters problems continuing the approximation near points of vertical tangency, the algorithm sets conditions which should stop the approximation prior to reaching such a point -- in the semicircles example above, this has the effect of keeping the graphs of the semicircles from quite reaching the x-axis.