c07 - 18.03 Class 7, Feb 22, 2006 Applications of C:...

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_________________________ 18.03 Class 7, Feb 22, 2006 Applications of C: Exponential and Sinusoidal input and output: Euler: Re e^{(a+bi)t} = e^{at} cos(bt) Im e^{(a+bi)t} = e^{at} sin(bt) [1] Integration Remember how to integrate e^{2t} cos(t) ? Use parts twice. Or: Differenting a complex valued function is done on the real and imaginary parts. Same for integrating. e^{2t} cos(t) = Re e^{(2+i)t} so int e^{2t} cos(t) dt = Re int e^{(2+i)t} dt and we can integrate exponentials because we know how to differentiate them! - int e^{(2+i)t} dt = (1/(2+i)) e^{(2+i)t} + c We need the real part. Expand everything out: 1/(2+i) = (2-i)/5 e^{(2+i)t} = e^{2t} (cos(t) + i sin(t)) so the real part of the product is (1/5) e^{2t} (2 cos(t) + sin(t)) + c More direct than the high school method! [2] Sinusoidal signals: Solve x' + 2x = cos(t) : toy model for a cooler responding to oscillating temperature. Use Variation of Parameter.
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This note was uploaded on 01/31/2011 for the course MAT 17A taught by Professor Staff during the Winter '08 term at UC Davis.

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c07 - 18.03 Class 7, Feb 22, 2006 Applications of C:...

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