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4/17/2011
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Example canonical ODEs
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Falling object
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Pendulum
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The Lorenz system
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Numerical solution of an ODE: general method
Lecture 19: Numerical solutions of ODEs
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Euler explicit scheme
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Euler implicit scheme
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Trapezoidal method
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Mid point method
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Built in Matlab functions
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Numerical error and instability
Example 1: falling object
Canonical example of an ODE: falling object
ODE system associated with it (Newton’s law):
Example 1: falling object
ODE system associated with it (Newton’s law):
This system can be integrated, which provides us with velocity
The velocity system can be integrated, which provides us with
trajectory information
This trajectory information solves the problem.
Example 2: pendulum equation
The pendulum equation for the system
reads:
Other canonical example of an ODE: pendulum equation
Example 2: pendulum equation
Pendulum equation:
With the standard approximation of small angles:
Leads to the standard harmonic oscillator:
With oscillation frequency:
Example 2: pendulum equation
In general, it is useful for numerical integration to write the ODE in
the form of a system of first order ODEs: start with the following:
Now with the following change of variables:
One obtains the following system (which is equivalent):
With the solution derived in the supplement lecture
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This note was uploaded on 09/07/2011 for the course ENGIN 7 taught by Professor Horowitz during the Spring '08 term at University of California, Berkeley.
 Spring '08
 HOROWITZ

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