I need a little explanation here the general solution is $$x(t)=c_1u(t)+c_2v(t)$$ where $u(t)=e^{\lambda t}(\textbf{a} \cos \mu t-\textbf{b} \sin \mu t$ and $v(t)=e^{\lambda t}(\textbf{a} \sin \mu t +\textbf{b} \cos \mu t)$ I am confused on what happened to the $i$ that is suppose to be in front of $v(t)$ and why it just "goes away". When they were deriving this formula they just go from writing it as $x^{1}(t)=u(t)+iv(t)$ to the general solution which is throwing me off as to what happened to the $i$. Nothing can just disappear in math without reason.
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$\begingroup$It doesn't really disappear.
Note that $\{u,v\}$ is linearly independent over $\mathbb R$, so if they are solutions of a second degree ordinary differential equation with constant coefficients, they form a basis of solutions.
The $i$ disappears because usually one is interested in real functions.Of course $u+iv$ will also be a solution to the differential equation, just not a real solution. There's probably a hidden assumption that you're only looking for real solutions.
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