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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 Winter '08
 Staff
 Linear Equations, Equations, Elementary algebra, e^

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