set3 - One Dimensional Systems Considerthe equation x fx t)...

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1 One – Dimensional Systems Consider the equation where x is a scalar. If only, then system is autonomous; otherwise it is non-autonomous. There are very few one-dimensional nonlinear equations that are exactly solvable. Consider the following examples: 1. Separable Systems Separable Systems : ! f(x, t) f(x) ! ! ! ! dx x f(x, t) f(x)g( t)or f(x)g( t) dt ! ! x f(x, t)
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2 By the method of separation of var iables, Integrals are solvable through quadrature (either exact closed form solutions in terms of special functions exist or the integrals can be evaluated by numer ical integration). a. As an example, consider the problem of a sphere falling in a viscous f luid: ! " " 0 0 x t x t dx or dx g(t)dt f(x) # ! dx g( t)dt f(x) g mg F drag v
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This is integrable by quadrature ! " " $ ! % % 2 drag StokesLaw Newton'sLaw ofresis tance ofresis tance a,b 0 F av bv ! % % ! 2 W ecan integratebyseparation ofvariables: v g (a/m)v (b/m)v ! % % ! 2 Theequation ofmotion isthen: mv mg av bv ! % % 2
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This note was uploaded on 12/29/2011 for the course ME 680 taught by Professor Na during the Fall '10 term at Purdue.

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set3 - One Dimensional Systems Considerthe equation x fx t)...

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