Are General Solutions of Linear ODEs Always Equivalent?

[Jhenrique]

Given the following ODE:

$ay''(t) + by'(t) + cy(t) = 0$

The following solution:

$y(t) = c_1 \exp(x_1 t) + c_2 \exp(x_2 t)$

is more general than:

$y(t) = A \exp(\sigma t) \cos(\omega t - \varphi)$

? Why?

 

[AlephZero]

The solutions are equivalent if $x_1$ and $x_2$ are complex conjugate numbers.

They are not equivalent if $x_1$ and $x_2$ are unequal real numbers, unless you want to use a crazy interpretation of $cos(\omega t - \varphi)$ where $\omega$ and $\varphi$ are complex constants.