Complex Solutions to Oscillations

[WWCY]

Homework Statement

Homework Equations

The Attempt at a Solution

I tried differentiating both sides of 3 and re-arranging it such that it started to look like equation 2, however i got stuck with 2 first order terms z' and couldn't find a way to manipulate it into a function z.

I then tried using a trial solution provided: z(t) = e^-iΩt and differentiating it according to what I derived above, before again, trying to rearrange them such that they would fit equation 2 had I subbed the trial solution into 2. I got stuck in both situations.

Any pointers and direction would be greatly appreciated

 

[Orodruin]

Please show us your actual attempt instead of vaguely describing it. If you do not show us what you got, we can only guess where you went wrong.

 

[haruspex]

i got stuck with 2 first order terms z'

So reuse eqn 3.

 

[WWCY]

So reuse eqn 3.

I gave it a shot, does this look right?

If this is unreadable I'll upload an image of better quality later as I am on public transport right now, apologies!

Please show us your actual attempt instead of vaguely describing it. If you do not show us what you got, we can only guess where you went wrong.

I made a ridiculous error in my previous solution, i'll do up another one using the same line of reasoning and upload it to see where I went wrong.

Thank you both for the help.