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
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.