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 May 21st, 2019, 05:10 PM #1 Senior Member   Joined: Dec 2015 From: somewhere Posts: 541 Thanks: 82 Differential equation $\displaystyle \frac{d^4 z}{dt^4 }=z+C \;$ , C-constant . To get at least one solution , apply derivatives to equallity : $\displaystyle z^{(5)}=z'$ , How to get the general solution ? Last edited by idontknow; May 21st, 2019 at 05:17 PM.
May 21st, 2019, 06:38 PM   #2
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 Originally Posted by idontknow $\displaystyle \frac{d^4 z}{dt^4 }=z+C \;$ , C-constant . To get at least one solution, apply derivatives to equality: $\displaystyle z^{(5)}=z'$. How to get the general solution?
To get the homogenous equation:
$\displaystyle \dfrac{d^4 z}{dt^4} - z = 0$

The characteristic equation is $\displaystyle m^4 - 1 = 0$, which has solutions 1, -1, i, -i. Thus the homogeneous solution is of the form:
$\displaystyle z_h(t) = Ae^t + Be^{-t} + D~\sin(t) + E~\cos(t)$.

Or, if you prefer
$\displaystyle z_h(t) = Ae^t + Be^{-1} + De^{it} + Ee^{-it}$
(The D's and E's are different from each other in the two equations.)

Can you finish?

-Dan

Last edited by skipjack; May 22nd, 2019 at 12:07 AM.

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