Write down the Lagrangian [1] of your system and deduce from it the associated Euler-Lagrange equation [2]. The sought-after frequencies are those of the normal modes [3] of this equation.

[1] https://en.wikipedia.org/wiki/Lagrangian_mechanics

[2] https://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation

[3] https://en.wikipedia.org/wiki/Normal_mode

Thank you Behnam Farid , I was able to get it by following the steps you highlighted

Emmanuel Anuoluwa Bamidele : My pleasure. Good to know that you have found the frequencies you were looking for.