Toronto Math Forum
MAT2442013F => MAT244 MathTests => Quiz 3 => Topic started by: Razak Pirani on November 07, 2013, 12:35:09 PM

4.2 #18 Find the general solution of the given differential equation.
y^{(6)}  y'' = 0
r^{6}  r^{2} = 0
r^{2}(r^{4}  1) = 0
r^{2}(r^{2}  1)(r^{2} + 1) = 0
r^{2}(r  1)(r + 1)(r  i)(r + i) = 0
r_{1,2} = 0
r_{3} = 1
r_{4} = 1
r_{5} = i
r_{6} = i
y(t) = c_{1} + c_{2}t + c_{3}e^{t} + c_{4}e^{t} + c_{5}cost + c_{6}sint

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^41) \implies r^2(r^2+1)(r^21) = 0 \implies r^2(r^2+1)(r1)(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}