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Calculus and Beyond Homework Help
Integration via complex exponential
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[QUOTE="tylersmith7690, post: 4495204, member: 485652"] [h2]Homework Statement [/h2] Using the complex exponential, nd the most general function f such that [itex]\frac{d^2f}{dt^2}[/itex] = e[SUP]-3t[/SUP] cos 2t , t all real numbers. [h2]Homework Equations[/h2] I'm having a lot of trouble with this question, my thinking is to integrate once and then one more time to undo the second derivative. However I am wondering if there is a trick to do this as i know there is a trick to doing it when say finding the 45th derivative of say e^t cos 5t. [h2]The Attempt at a Solution[/h2] d/dt = ∫ e[SUP]-3t[/SUP] cos (2t) dt = e[SUP]-3t[/SUP] . Re(e[SUP]2ti[/SUP] = Re ∫ e[SUP](-3+2i)[t/SUP] = Re[ 1/(-3+2i) . e[SUP](-3+2i)t[/SUP] ] + C = Re [ 1/(-3+2i) . (-3-2i/-3-2i) . e[SUP](-3+2i)t[/SUP] ] + C = Re [ [itex]\frac{-3-2i}{13}[/itex] . e[SUP]-3(cos2t+isin2t)[/SUP] = Re [ [itex]\frac{e^-3t}{13}[/itex] .( -3 cost t + sin 2t - i(3sin 2t + 2 cos 2t) +C = [itex]\frac{e^-3t}{13}[/itex](-3 cos 2t + sin 2t) +C What is next to do or is there a trick in an earlier step?[/SUP] [/QUOTE]
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Integration via complex exponential
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