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Original post by Teenage Hype
Can anyone explain #2 to me please? Not sure how to get from the second last line to the last line

http://www.ctralie.com/Teaching/Euler/


For f(x)=af(x)f'(x)=af(x) it must be true that f(x)=eaxf(x)=e^{ax} (can be shown via solving the differential equation y=ayy'=ay such that y(0)=1y(0)=1) and since f(x)=cos(x)+isin(x)f(x)=\cos(x)+i\sin(x) we get:

f(x)=eax=cos(x)+isin(x)f(x)=e^{ax}=\cos(x)+i\sin(x) where a=1=ia=\sqrt{-1}=i
(edited 7 years ago)
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