I've been given the following problem:
Solve the initial value problem $y^{(6)} - 3y^{(4)} + 3y^{(2)} - y = 0$ with $y_0 = 0, y_1 = 1, y_2 = -1, y_3 = 0, y_4 = 0$ and $y_5 = 0$ at $t = 0$.
I've been able to formulate the characteristic polynomial: $\lambda^6 - 3 \lambda^4 + 3 \lambda^2 - \lambda = 0$, however I can't seem to be able to factor it in any way that'll allow me to solve for the roots. I presume there's a simple trick to do so, I just don't see it.
Please note that this is a homework problem, so I'm not looking for the solution, just some insight into how to go about solving for the roots of the characteristic polynomial.
Thanks in advance!
You're characteristic polynomial has a mistake. Note that $y^{(k)}$ gives $\lambda^k$, so $y$ (i.e. $y$ with $0$ derivatives) will give $\lambda^0 =1$. Thus you should get $$\lambda^6 - 3\lambda^4 + 3\lambda^2 - 1 = 0.$$
Now for a hint: what is $(x-1)^3$?