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May 21st, 2019, 06:10 PM   #1
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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.
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May 21st, 2019, 07:38 PM   #2
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Originally Posted by idontknow View Post
$\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?

Thanks from idontknow and Greens

Last edited by skipjack; May 22nd, 2019 at 01:07 AM.
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