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Non-Linear Differential Equations - First Integrals

Hey,
Any help on the attached question would be most appreciated. Particularly b) - e).
No idea what I'm doing really and can't find much on the internet about first integrals that isn't beyond my level.
Cheers!
First integrals are essentially just constants of the motion, so to show that I (say) is a first integral all you have to do is show that dI/dt=0.

And btw that isn't a non-linear DE, technically it's a linear 4th order ODE.

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