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Can someone give me hint on how to show that the fact that $\alpha$ is a root of order $l+1$ of the polynomial $p(x)=x^n+a_1x^{n-1}+\cdots+a_n$ implies that $t\mapsto y^l e^{\lambda y}$ is a solution of the ODE $x^{(n)}+a_1 x^{(n-1)}+\cdots + a_n x=0$ ?

I tried doing a direct calculation, but didn't succeed (altough I would prefer this direct route, since I don't yet see all the connection between the different theorems of the theory for linear systems of ODEs, so it would be easier for me to understand)...then I tried using some theory about ODEs but didn't get far: The ODE is equivalent to a system of linear ODEs of order 1 and the characteristic polynomial of the matrix of that system is precisely the polynomial above, but there I kind of stop dead, since calculating the Jordan normal form is just horrible, so I thought there must be some easier, less calculation-heavy way. Could you show me the way ?

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The most succinct way I know how to do this is to think about differential operators. Letting $D$ denote the differentiation operator, we want to compute the nullspace of $p(D)$ where $p$ is some polynomial. The first basic observation is that if $p(z) = \prod (z - \lambda_i)^{m_i}$, then $$p(D) = \prod (D - \lambda_i)^{m_i}$$

and each of the operators in this product commute with each other; in particular, if $(D - \lambda_i)^{m_i} f = 0$ for some function $f$, then $p(D) f = 0$ as well. So we are reduced to the problem of solving $$(D - \lambda_i)^{m_i} f = 0.$$

Since we know that $(D - \lambda_i) f = 0$ has solution $f = C e^{\lambda_i x}$, it might occur to us to write $f = e^{\lambda_i x} g$ where $g$ is some function.

Proposition: $(D - \lambda_i)^{m_i} e^{\lambda_i x} g = e^{\lambda_i x} D^{m_i} g$.

Proof. It suffices to compute that $(D - \lambda_i) e^{\lambda_i x} g = Dg$.

It follows that $(D - \lambda_i)^{m_i} e^{\lambda_i x} g = 0$ if and only if $D^{m_i} g = 0$, hence if and only if $g$ is a polynomial of degree at most $m_i - 1$.

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Define a series of variables: $x_1 = x,x_2=\dot x, x_3= \ddot x, \ldots$. Then, rephrase the given ODE as $$ x^{(n)}= -a_1 x^{(n-1)}+\cdots - a_n x $$ This allows us to write the differential equation in a compact matrix equation form $$ \dot x(t) := A x(t) = \pmatrix{0 &1 &0 &\ldots &0\\0 &0 &1 &\ldots& 0\\ \vdots &&\ddots &\ddots &0\\0 &&&0&1\\-a_n &-a_{n-1} &\ldots &-a_2 &-a_1}x(t) $$

Then, the solution of this matrix differential equation is given by $$ x(t) = e^{At}x_0 = V^{-1}e^{Jt}Vx_0 $$ The determinant of $(x I-A)$ is $p(x)=x^n+a_1x^{n-1}+\cdots+a_n$ and can be shown by expanding along the last column. Then, you don't need to calculate much. The remaining step is assuming an arbitrary sized Jordan block in $J$ and then computing e^{Jt}. There you will see that when $\alpha$ is an eigenvalue of $A$ associated with a Jordan block of size at least two then, the result follows. Look at the state-transition matrix in this link for the general formula.

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Yes, but how does the result follow from looking $e^{Jt}$ ? All I get from that is the components of $x(t)$ are linear combinations of $e^{\alpha t} t^l$, with $l\geq 1$, if the Jordanblock has at least size two. I haven't used, as far as I can see, at any point that $\alpha$ is a root of order $l+1$ of the polynomial $p$. Or am I maybe missing something ? –  user19822 Dec 5 '11 at 7:11
@user19822 Set $t=y$ and $\alpha=\lambda$ and that seems about it. –  user13838 Dec 5 '11 at 11:49
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