Differential Equations Ordinary and Partial Differential Equations Math Forum

 May 21st, 2019, 06:10 PM #1 Senior Member   Joined: Dec 2015 From: Earth Posts: 825 Thanks: 113 Math Focus: Elementary Math 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 06:17 PM. May 21st, 2019, 07:38 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 01:07 AM. Tags differential, equation Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Tamer Ghareeb Differential Equations 5 December 7th, 2015 01:30 AM Sonprelis Calculus 6 August 6th, 2014 11:07 AM PhizKid Differential Equations 0 February 24th, 2013 11:30 AM Vasily Differential Equations 4 June 4th, 2012 03:17 AM main Differential Equations 1 July 3rd, 2009 10:52 AM

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