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Ordinary differential equations
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ODE's
Many dynamical systems in physics, astronomy, chemistry, physiology, meteorology, economics, population
dynamics can be described by
Ordinary Differential Equations.
In the second half of the 20th century much attention has been focussed on the
often
chaotic,
i.e. unpredictable behaviour of (nonlinear) ODE's. A wellknown example is the
Lorenz atrractor, illustrating the "Butterfly effect": small causes can have large effects.
Mathgrapher uses an accurate AdamsBashforth variable order, variable step predictorcorrector
algorithm to integrate systems of up to 20 coupled differential equations.
Several analytical tools
are available for ODE's such as:
Examples:
ODE's  Introduction
An ordinary differential equation is a differential equation that contains
only one variable, for instance Time. Such an equation may contain derivatives
and derivatives of derivatives. The highest derivative determines the order of the ODE.
For example, Newton's law for the acceleration of a particle in a gravitational
field (in one dimension) is described by a second order ODE:
Many dynamical systems can be discribed by such an higher order ODE.
The above 2nd order equation is equivalent to the following set of first order
ODE's
This example is an example of the following general rule: Any nth order ordinary differential
equation can be reduced to a set of n first order differential equations.
Mathgrapher allows you to integrate a set of 20 coupled firstorder equations.
So, first you have to write your differential equation in an equivalent set of
first order equations as in the example given above. In the example given
above you would define F1=x(t), F2=v(t) and dF1/dt=F2 , dF2/dt=c/x^2. Set the
initial values of F1 and F2, the initial and final values of the time and start
the integration. When the integration is finished you may draw the orbit.
It is straightforward to extend this example to 2 or 3 dimensions and
calculate the orbits of a particle in a central force (gravitational) field,
and to add frictional terms to study, for example, the orbit of a satellite
in the atmosphere of the earth.
