(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the general solution to

y'''+y=0

2. Relevant equations

3. The attempt at a solution

y''+y=0

r^3+1=0

r^3=-1

r=(-1)^(1/3)

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-1=-1+0i

-1=cos(pi)+isin(pi)=e^(i*pi)

-1=cos(pi+2xpi)+isin(pi+2xpi)=e^i(pi+2xpi)

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(-1)^(1/3)=(e^i(pi+2xpi))^(1/3)

(-1)^(1/3)=(e^i(pi/3+2xpi/3))

(-1)=cos(pi/3+2xpi/3)+isin(pi/3+2xpi/3)

-----------------------------

How do I get this to the general solution form? I know I do something where I let x=0,1,2,3,etc.

But I am not on sure what steps to take.

The answer on Wolfram Alpha is

"y(x) = c_1 e^(-x)+c_2 e^(x/2) sin((sqrt(3) x)/2)+c_3 e^(x/2) cos((sqrt(3) x)/2)"

Thanks in advance!

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