### Author Topic: Problem 1 (Day section)  (Read 3396 times)

#### Razak Pirani

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• • Posts: 3
• Karma: 5 ##### Problem 1 (Day section)
« on: November 07, 2013, 12:35:09 PM »
4.2 #18 Find the general solution of the given differential equation.

y(6) - y'' = 0

r6 - r2 = 0
r2(r4 - 1) = 0
r2(r2 - 1)(r2 + 1) = 0
r2(r - 1)(r + 1)(r - i)(r + i) = 0

r1,2 = 0
r3 = 1
r4 = -1
r5 = i
r6 = -i

y(t) = c1 + c2t + c3et + c4e-t + c5cost + c6sint
« Last Edit: November 07, 2013, 12:37:37 PM by Razak Pirani »

#### Mark Kazakevich

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• • Posts: 4
• Karma: 2 ##### Re: Problem 1 (Day section)
« Reply #1 on: November 07, 2013, 12:36:03 PM »
For the differential equation:
\begin{equation} y^{(6)} - y'' \end{equation}

We assume that $y = e^{rt}$.
Therefore, we must solve the characteristic equation:

\begin{equation} r^6 - r^2 = 0 \end{equation}

We find:
$r^6 - r^2 = 0 \implies r^2(r^4-1) \implies r^2(r^2+1)(r^2-1) = 0 \implies r^2(r^2+1)(r-1)(r+1) = 0$

This means the roots of this equation are:

$r_1 = 0, r_2=0, r_3=i, r_4=-i, r_5=1,r_6=-1$
(We have a repeated root at r = 0)

So the general solution to (1) is:
\begin{equation} y(t) = c_1 + c_2t + c_3\cos{t} + c_4\sin{t} + c_5e^{t} + c_6e^{-t} \end{equation}