Question

What are the roots of this equation :( Z'' means (d^2Z/dX^2))

Z''''+Z''+1=0

Answer 1

This is a differential equation. We don't "find the roots of a differential equation."
Instead, we write an "auxiliary equation" and find the roots of that.

Such an aux. eqn. would be m^4-m^2+1=0.

I'd choose some dummy variable, such as p, and let p=m^2.

Then our aux. eqn. would become p^2+p+1=0.
Find the roots of that (they are complex).

I've found that \[p=\frac{ -1 plusorminus \sqrt{-3} }{ 2 }=m ^{2}.\]
Unfortunately, that's going to be difficult to solve for m!

I'm afraid i'm going to have to defer to someone else in solving this problem. Good luck!