I'm having trouble wrapping my head around the following issue. My book solves a problem without using complex exponential solution like $C_1 e^{it}$ and using either $A \cos(t) + B \sin(t)$ or $A \cos(t+\phi)$ to a second-order "undamped" linear ODE. I understand that the solution can be purely real or complex. The trouble arises when I use the solution to this ODE inside another function which cubes this solution. Cubing $C_1 e^{it}$ results in another complex sinusoid with 3 times the frequency, which is a very different effect from cubing something like $A \cos(t+\phi)$
Here is my work. I've highlighted where I think things have gone awry.
The complex exponential solutions make sense for linear ODEs because you can get a characteristic polynomial since the exponential is the eigenfunction of the derivative operator. The reason why a complex solution times any constant is still a complex solution is due to the linearity in the linear ODE. If $x(t)$ is a solution and $i*y(t)$ is a solution then any linear combination is also a solution. Additivity and homogeneity are not properties of the solutions of a non-linear ODE and so it makes no sense to assume that the solution is of the form $e^{st}$ .