c03 - 18.03 Class 3, Feb 13, 2006 First order linear...

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18.03 Class 3, Feb 13, 2006 First order linear equations: Models Vocabulary: Coupling constant, system, signal, system response, Models: banks, mixing, cooling, growth and decay. Solution in case the equation is separable; general story deferred to Class 4. [1] If I had to name the most important general class of differential equations it would be "linear equations." They will occupy most of this course. Today we look at models giving first order linear equations. Definition: A "linear ODE" is one that can be put in the "standard form" ___________________________ | | | x' + p(t)x = q(t) |___________________________| | When t = time is the independent variable, the notation x-dot is often used. In these notes I'll continue to write x' however. [2] Model 1. Bank account: I have a bank account. It has x dollars in it. x is a function of time. I can add money to the bank and make withdrawals. The bank is a system. It pays me for the money I deposit! This is called interest. In the old days a bank would pay interest monthly: Then Delta t = 1/12 and x(t + Delta t ) = x(t) + I x(t) Delta t [ + . ... ] I has units (year)^{-1} . These days I is typically about 2% = 0.02 . You don't get 2% each month! you get 1/12 of that.
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c03 - 18.03 Class 3, Feb 13, 2006 First order linear...

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