Systems of Differential Equations
Graphs solution functions
x(t) and
y(t) to the
system of differential equations
x '=
f(
x,
y) and
y '=
g(
x,
y), with initial values given by
x(
t_{0})=
x_{0} and
y(
t_{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) for x' in the text input field marked
"x'=f (x,y)="
(Example: x'=y)
- Enter the function
g(x,y) for y' in the text input field marked
"y'=g(x,y)="
(Example: y'=x)
- Click the "Graph" button
(this button also refreshes the graph)
- The initial values x_{0}, y_{0},
and t_{0} can be changed in either of two ways:
- Enter the values directly into the text input fields marked
"x_{0}=",
"y_{0}=", and
"t_{0}="
- Click and drag the red points on the graph --
horizontally, the red points move together,
and their horizontal position
corresponds to the value of t_{0}; vertically, the points move
independently, with the vertical position of the point on the
blue graph
corresponding to the value of x_{0},
and the vertical position of the point
on the green graph corresponding to
y_{0}
(Example: x_{0}=0, y_{0}=1, and
t_{0}=0)
- To erase the graph and the text input fields for
f (x,y) and g(x,y), and
set x_{0}, y_{0}, and
t_{0} to default values, click the
"Clear" button
The text input fields for f (x,y) and
g(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 for x_{0}, y_{0}
and t_{0} can accept real numbers
in decimal notation.
Examples
Other Notes
The graph shows solution functions x(t) andy(t) to the
system of differential equations
x'=f (x,y) and
y'=g(x,y), with initial values given by
x(t_{0})=x_{0} and
y(t_{0})=y_{0}.
The particular solution functions x(t) and y(t)
to the system of differential equations satisfying the given initial values
will be graphed in blue (for x(t))
and green (for y(t)).
Since the functions f (x,y) and
g(x,y) do not depend on the
variable t, changes in the initial value t_{0}
only have the effect of horizontally shifting the graphs.
The particular solution functions x(t) and y(t)
are graphed by determining
numerical approximations to the functions using the classical (order four)
Runge-Kutta Method.
In the "Predator-Prey Model" example above
the particular
solution functions represent populations in a simple
two-species predator-prey
model. The
function x(t) (in blue) represents
the prey population, and y(t) (in green)
represents the predator population -- with the given initial values
x_{0}=3 and y_{0}=1, note the periodic
population explode-collapse pattern, with the predator population lagging
the prey population (with prey population high and predator population low,
prey population can grow, but then plentiful food supply makes predator
population also grow; overpredation causes prey population to collapse,
resulting in starvation and collapse for the predator population; low
predator population allows the prey population to recover, and the cycle
repeats). The two populations do have
equilibrium values, associated
with x_{0}=1 and y_{0}=1 at time
t_{0}. It is also
instructive to look at the
phase portrait for this system.