### Differential Equations and Initial Value Problems

Graphs solution functions y(x) to the differential equation y'=f (x,y), with initial value given by y(x0)=y0.
How to use   ||   Examples   ||   Other Notes

How to Use
• Enter the function f (x,y) in the text input field marked "y'=f (x,y)=" Example: y'=y)
• Click the "Graph" button (this button also refreshes the graph)
• Choose the initial value point (x0,y0) (Example: (x0,y0)=(0,1))
This can be done either of two ways:
• Enter the values directly into the text fields marked "x0=" and "y0=" and click the "Graph" button to refresh
• Use the mouse to click and drag the red point on the graph.
• To erase the graph and all input fields, click the "Clear" button
The text input field marked "y'=f (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 marked "x0=" and "y0=" can accept any decimal numbers.
Examples
 Function of x only: y'=cos(x) Exponential Growth: y'=y Exponential Decay: y'=-y Logistics Growth: y'=y(1-y) Semicircles: y'=-x/y Radial Lines: y'=y/x

Other Notes
The particular solution function y(x) to the differential equation satisfying the given initial values will be graphed in blue.

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.