Confusion about two modes of coupled oscillators having same frequency

[PrathameshR]

I am studying coupled oscillations and one of the refrance I'm using says that two modes can have same frequency whereas the other one says it's impossible to have same frequency for two modes. Please help.

 

[Dr.D]

For two modes to have the same natural frequency there must be repeated roots for the characteristic equation. This can happen although it is not very common in my experience.

 

[marcusl]

The example in the first book seems to be a special case where the x,y modes considered are independent of each other—the coupling is zero. Confirmation of that would be on previous pages that we can’t see.