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

2. Relevant equations

euler

##e^{ix} = cos(x) + i*sin(x)##

##e^{-ix} = cos(x) - i*sin(x)##

3. The attempt at a solution

I'm starting with differential equations and I'm trying to understand this solution including complex numbers:

First we determine the zeros. I understand that part. we get ##\lambda_1 = -i## and ##\lambda_2 = i## both with multiplicity 2.

So we get ##y_1(x)=e^{ix}## , ##y_2(x)=x*e^{ix}## , ##y_3(x)=e^{-ix}## and ##y_4(x)=x*e^{-ix}##. I'm good so far.

But now we consider the real combinations given by ##y_1(x)=cos(x)## , ##y_2(x)=x*cos(x)##, ##y_3(x)=sin(x)## , ##y_1(x)=x*sin(x)##

How do you obtain those real combinations? I see that ##Re(e^{ix})=cos(x)## but the sinus parts? And why are we allowed to do that? I can understand that in physics we can focus on the real solution but this is a strict math problem

Thanks in advance

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# Differential equation

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