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M334Lec29

# M334Lec29 - Math 334 Lecture#29 7.1,7.2,7.3 Introduction to...

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Math 334 Lecture #29 § 7.1,7.2,7.3: Introduction to Systems of First Order ODEs Example. (Converting a second order ODE into a system of two first order ODEs.) An IVP for a mass-spring system is mu + γu + ku = F ( t ) u (0) = u 0 , u (0) = u 0 . Define two new dependent variables x 1 = u and x 2 = u . The derivatives of these new dependent variables with respect to the independent variable t are x 1 = u = x 2 , x 2 = u = F ( t ) - γu - ku m = F ( t ) - γx 2 - kx 1 m . [The original second order ODE in u was solved for u and substituted into the second first order ODE.] In matrix form, this system of first order ODEs is x 1 x 2 = 0 1 - k/m - γ/m x 1 x 2 + 0 F ( t ) /m . This conversion of a second order ODE into a system of first order ODEs illustrates how to convert of an n th order ODE into a system of n first order ODEs. Example. (Obtaining systems of first order ODEs through modeling.) A dissolved chemical is mixed in a coupled system of two tanks, each of which contains 100 gallons of water.

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M334Lec29 - Math 334 Lecture#29 7.1,7.2,7.3 Introduction to...

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