18.03 Lecture Notes - Lecture 4: System Of Linear Equations, Integrating Factor, Flashcard
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Definition: a "linear ode" is one that can be put in the "standard form" | r(t)x" + p(t)x = q(t) | x = x(t) |_______________________________| r(t), p(t) are the "coefficients" [i may have called q(t) also a coefficient also on monday; this is not correct, fix it if i did. ] The left hand side represents the "system," and the right hand side arises from an "input signal. " A solution x(t) is a "system response" or "output signal. " We can always divide through by r(t), to get an equation of the. | x" + p(t)x = q(t) | x = x(t) (*) The equation is "homogeneous" if q is the "null signal," q(t) = 0 . This corresponds to letting the system evolve in isolation: In the bank example, no deposits and no withdrawals. In the rc example, the power source is not providing any voltage increase. The homogeneous linear equation x" + p(t) x = 0 is separable.
