Congratulations, you have just derived the modes of oscillation present in this coupled oscillator system! Allow us to explore what you have done in this question. The primary leap of faith was done in 4.2, where we guessed the form of the solution in order to proceed. From physical intuition, we could argue that if we choose the initial conditions carefully, we might be able to find the system in a single oscillation frequency as we only have springs and masses connected up without any other moving parts. Therefore we set both \(\omega\) in the assumed solution to be the same (but of course, if we run into contradictions later on then we should realise we were being too simplistic!).  
As a general rule of thumb, the number of modes present in a one-dimensional system is determined by the number of coupled masses present. Note that in 4.3 you have derived a mode where \(\omega=0\). This corresponds to the case where both masses move together with constant velocity, resulting in a "trivial" oscillation.

4.4 Create your own three mass system and solve accordingly.