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
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Examples
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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
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.