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## Homework Statement

Using the complex exponential, nd the most general function f such that

[itex]\frac{d^2f}{dt^2}[/itex] = e

^{-3t}cos 2t , t all real numbers.

## Homework Equations

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 im 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.

## The Attempt at a Solution

d/dt = ∫ e

^{-3t}cos (2t) dt

= e

^{-3t}. Re(e

^{2ti}

= Re ∫ e

^{(-3+2i)[t/SUP] = Re[ 1/(-3+2i) . e(-3+2i)t ] + C = Re [ 1/(-3+2i) . (-3-2i/-3-2i) . e(-3+2i)t ] + C = Re [ [itex]\frac{-3-2i}{13}[/itex] . e-3(cos2t+isin2t) = 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?}