### 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*(

*x*_{0})=

*y*_{0}.

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
(
*x*_{0},*y*_{0})
(Example:
(*x*_{0},*y*_{0})=(0,1))

This can be done either of two ways:
- Enter the values directly into the text fields marked
"
*x*_{0}=" and
"*y*_{0}="
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 "*x*_{0}=" and
"*y*_{0}=" 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.