Is there a way to find out the roots of the system numerically?
For the toy model described by the coupled second order differential equation (CoSODE),
the solution is,
Substituting the solutions back into the differential equations and simplifying yields
and multiplying the above together yields,
$F(w) = (aw^2)^2-c^2.$
Solving for roots when F(w) = 0 yields,
The above is a simple problem enough to allow us to solve it analytically and use it to find (possible complex) roots of F(w). Can the above be done to find roots of F(w) numerically with one undefined parameter say $a$ is unknown? The reason I am asking is that some CoSODE are more complicated and nonlinear hence finding the solution of the differential equation analytically/symbolically is impossible. Furthermore, the determinant cannot be employed in complicated case because of nonlinear terms. (Note that I do not have access to the latest version of mathematica (12)).