MIT18_03S10_c28

MIT18_03S10_c28 - 18.03 Class 28 Laplace Transform III...

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18.03 Class 28 , Apr 14, 2010 Laplace Transform III: Second order equations; completing the square. 1. Question period: f', FTC 2. L[f'_r(t)] 3. s-derivative rule 4. Second order equations Question period: [1] There were good questions from the class about finding the generalized derivative of f(t) = u(t) cos(omega t) . Let's do this again. The graph shows that f(t) is continuous except at t = 0 . The regular part of the derivative is f'_r(t) = - omega u(t) sin(omega t) and the singular part is f'_s(t) = delta(t) You can also apply the product rule: f'(t) = - omega u(t) sin(omega t) + delta(t) cos(omega t) = - omega u(t) sin(omega t) + delta(t) since cos(omega 0) = 1 . In order for the "fundamental theorem of calculus" integral_a^c f'(t) dt = f(c) - f(a) to be true, you MUST use the generalized derivative. For example, take a = -1, b = 1, f(t) = u(t) . If you say u'(t) = 0 , then integral_a^c u'(t) dt = 0 , not 1 ... while integral_a^c delta(t) dt = 1 [2] At the end of class on Monday I took a regular function f(t) (with f(0) = 0 for t < 0 ) whose only jump in value is at t = 0 , and found L[f'_r(t)] : f'(t) = f'_r(t) + f'_s(t)
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