supplement_ODE_6p

supplement_ODE_6p - Supplement lecture ODEs What is an ODE...

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4/14/2011 1 What is an ODE? Where do ODEs come from? The harmonic oscillator Damped variations The Lorenz attractor Supplement lecture: ODEs What is an ordinary differential equation? An ordinary differential equation is an algebraic relation which relates a function with its derivatives. For example, for a function f, the following equation is an ordinary differential equation: f’(x) +f(x) = 0 Just like when you had a quadratic equation, for example x+1 = 0 you needed to find the solution (x=-1), here you will need to find the solution, i.e. f(x), for all x What is an ordinary differential equation? Example of a first order ODE Solution method: separation of variables: Solution is an exponential Where do ODEs come from? In numerous engineering problems, the constitutive equations result from the laws of physics, and are given in the form of ODEs For example: a spring: Newton’s law: Leads to the following ODE Example of solution of the harmonic oscillator Harmonic oscillator
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