What are the complex solutions of a linear homogenous ODE of order $n$ with constant coefficients? What are the complex solutions of a linear homogenous ODE of order $n$ with constant coefficients? Where can I read a proof?

p.s. I don't even see the answer to the first question with a google search.
Also you can consult the following textbook:
L. Perko, "Differential Equations and Dynamical Systems," chapter 1.
The wikipedia page entitled Homogenous Differential Equations gives the method right at the start; a roof (depending on the degree of rigor required) is straightforward because the method is fairly simple.