MIT18_03S10_c09

MIT18_03S10_c09 - 18.03 Class 9 Linear vs Nonlinear a...

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Linear vs Nonlinear: a debate [1] Linearization near equilibrium [2] Exponential Response Formula (first order) [3] Potential blow-up of solutions to a nonlinear equation Hour Exam I Wednesday. Locations for both lectures: 10-250 A -- G Walker H -- Z Office hours as usual on Wednesday Review: Nonlinear vs Linear y'(t) = F(t,y(t)) vs r(t) x'(t) + p(t) x(t) = q(t) This is in the form of a debate, between Linn E. R. (on the right) and Chao S. (on the left). I'll take a vote at the end. Linn: I'd like to begin by making the point that there is a solution procedure for linear equations, which reduces solution of any linear equation to integration: Multiply the equation through by a factor so that the two terms become the two terms in (d/dx)(ux) . Then integrate. Sometimes you can just *see* this: t^2 x' + 2t x = (d/dt)(t^2 x) for example. If we are in *reduced* standard form, so r = 1 , then this can be done systematically: We seek u(t) such that u (x' + px) = (d/dt) (ux) i.e. pu = u' : separable, with solution u = e^{\int p(t) dt} (any constant of integration will do here). Then integrate both sides of (d/dt) (ux) = uq : x = u^{-t} int uq dt The constant of integration is in this integral, so the general solution has the form x = x_p + c u^{-1} . Another lovely feature of linear equations is that the constant of integration in the solution of a linear equation always appears right there. The "associated homogeneous equation" is x' + px = 0 : separable, with solution x_h = e^{-\int p(t) dt} . Look! this is the reciprocal of the integrating factor! Wonderful! In most applications, u{-1} falls off to zero as t gets large; the term c u^{-1} is a "transient." Chao: That's a lot of integration . ... I'm more interested in the general

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MIT18_03S10_c09 - 18.03 Class 9 Linear vs Nonlinear a...

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