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Homework Statement
Homework Equations
euler
##e^{ix} = cos(x) + i*sin(x)##
##e^{ix} = cos(x)  i*sin(x)##
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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