Problem 1 (Day section)

[Razak Pirani]

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

r_1,2 = 0
r₃ = 1
r₄ = -1
r₅ = i
r₆ = -i

y(t) = c₁ + c₂t + c₃e^t + c₄e^-t + c₅cost + c₆sint

[Mark Kazakevich]

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}