# How can the method of complex exponentials be used to find a complex solution i(t) which varies harmonically in time.

I am WEAK in DE's. I have been trying to understand this question all day. I posted the question as an image since I don't have enough rep points to post it here.

http://images.4chan.org/sci/src/1327448267650.jpg

There is one example in my text Butkov on how to do this but it is not clear at all to me what I must do!

so far I just wrote:

L* i"(t) + R* i'(t) + (1/C) i(t) = -Vo*ω*sin(ω*t)

I then took a function f(t) = F*exp(i*t)

where F = Vo*ω, and so F*exp(i*t) = Vo*ω(cosωt - i*sinωt)

I then wrote im{Vo*ω*exp(-it)} = -Voωsinωt

I clearly have no idea what I am doing. Please, any help would be appreciated.

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The link isn't working for me. – Gerry Myerson Jan 24 '12 at 23:47
My apologies. I reposted it on Photobucket. Perhaps this website restricts certain urls. s560.photobucket.com/albums/ss45/SnoopyRedBaron/… – user23463 Jan 24 '12 at 23:59

## 1 Answer

You're trying to solve the differential equation $L i''(t) + R i'(t) + (1/C) i(t) = - V_o \omega \sin(\omega t)$. The right side is the imaginary part of $- V_o \omega e^{i\omega t}$, so the idea is to first find a solution of the differential equation with right side replaced by $-V_o \omega e^{i\omega t}$, and the imaginary part of that solution will be a solution of your differential equation. Try a solution of the form $y(t) = A e^{i\omega t}$: plug this in to the (new) differential equation, and see if you can find a complex constant $A$ that makes the equation true.

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