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Eliminate

Here are two slightly unusual applications of Eliminate.

Integro-Differential Equation

Consider the integro-differential equation

[Graphics:../Images/index_gr_114.gif]

Mathematica has no built-in function for solving such equations. Jürgen Tischer (jtischer@col2.telecom.com.co) suggested the following elegant approach. Differentiating with respect to [Graphics:../Images/index_gr_115.gif]

[Graphics:../Images/index_gr_116.gif]
[Graphics:../Images/index_gr_117.gif]

and eliminating the integral yields a trivial differential equation.

[Graphics:../Images/index_gr_118.gif]
[Graphics:../Images/index_gr_119.gif]

The solution to this equation for the pair of initial conditions, [Graphics:../Images/index_gr_120.gif] and [Graphics:../Images/index_gr_121.gif], is immediate.

[Graphics:../Images/index_gr_122.gif]
[Graphics:../Images/index_gr_123.gif]

It is easy to verify that this solution satisfies the integro-differential equation.

[Graphics:../Images/index_gr_124.gif]
[Graphics:../Images/index_gr_125.gif]

System of Differential Equations

Consider a system of differential equations.

[Graphics:../Images/index_gr_126.gif]

It should be apparent that [Graphics:../Images/index_gr_127.gif] and [Graphics:../Images/index_gr_128.gif] can be eliminated from this linear system of equations. Jens-Peer Kuska (kuska@linmpi.mpg.de) suggested an elegant method for doing this using Eliminate. After differentiating the system of equations,

[Graphics:../Images/index_gr_129.gif]
[Graphics:../Images/index_gr_130.gif]

we can then eliminate [Graphics:../Images/index_gr_131.gif], [Graphics:../Images/index_gr_132.gif], and [Graphics:../Images/index_gr_133.gif].

[Graphics:../Images/index_gr_134.gif]
[Graphics:../Images/index_gr_135.gif]


Converted by Mathematica      September 30, 1999

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