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c09 - 18.03 Class 9 Review Linear v Nonlinear[1 review of...

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18.03 Class 9, Feb 27, 2006 Review: Linear v Nonlinear [1] review of linear methods [2] Comment on special features of solutions of linear first order ODEs not shared by nonlinear equations. [1] First Order Linear: x' + p(t) x = q(t) system; input signal; output signal = system response. General comment: for first order LINEAR equations, the solutions always are of the form x = x_p + c x_h where x_p is SOME solution ("particular solution") and x_h is a nonzero solution of the homogeneous equation. If p > 0 , c x_h deserves to be called a "transient"; it dies away and leaves x_p. Decision tree for solving first order linear equations Separable? ( p and q are both constant, or if q = 0 ) - If yes, then solve by separation of variables. - If no: Is the "coefficient" p constant?: "constant coefficient" - If yes, solution to homogeneous equation is e^{-pt} . - Is the signal exponential? q(t) = B e^{rt} , r constant If so, try x = A e^{rt} and solve for A. - Is the signal sinusoidal? especially B cos(omega t) If so, replace with z' + p z = B e^{i omega t} , solve that, and take the real part of the solution.

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c09 - 18.03 Class 9 Review Linear v Nonlinear[1 review of...

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