linear ODE with constant coefficients, proof Let $
\sum\limits_{k = 0}^n {y^{\left( k \right)} a_k }  = 0
$ an homogeneous ODE, where $a_k$ are constants. How can I solve the equation when the roots are repeated?  One way, that I saw in wikipedia, is using the fact that if $ e^{cx} $ is a solution, then $ (x^r)(e^{cx}) $ is also too, How can I prove this? It´s difficult to me, to evaluate the sum, because i want to show that $
\sum\limits_{k = 0}^n {\left( {x^r e^{cx} } \right)^{\left( k \right)} a_k }  = 0
$ but I need to evaluate $
{\left( {x^r e^{cx} } \right)^{\left( k \right)} }
$ and I don´t know how to do it Dx

I don't know if you have the background for this, but I do not think there is much choice if you allow the degree  $n \geq 3.$ It is not so bad for $n=2.$
Anyway, you introduce a bunch of variables $y_0 = y, y_1 = y', y_2 = y'',$ and so on. As your coefficients are constant, we may divide through by whatever $a_n$ might be to arrive at a revised constant coefficient equation with $a_n = 1.$ 
So the new system of equations is a linear system beginning with $$y_0' = y_1, \; y_1' = y_2, \ldots, \; y_{n-2}' = y_{n-1},$$ but finally 
$$ y_{n-1}' = - b_0 y_0 - b_1 y_1 - \ldots - b_{n-1} y_{n-1},$$ where I have taken $b_j = a_j / a_n.$  
We write a column vector $Y$ with entries $y_0, y_1, \ldots, y_{n-1}.$ that system now becomes
$$  Y' = B Y $$ with initial conditions written as
$$ Y(0) = Y_0$$ 
and the solution to the system is
$$ Y = e^{B x} Y_0$$ 
Note that $B$ has exactly the form of a companion matrix, see the square matrix at
http://en.wikipedia.org/wiki/Companion_matrix#Linear_recursive_sequences
The appearance of $x, x^2,\ldots$ comes from the Jordan normal form of $B,$ precisely when there are repeated eigenvalues (characteristic values) of $B,$ and when there are off-diagonal entries in the relevant Jordan block. If there are repeated roots but the Jordan normal form is diagonal anyway, then no polynomial terms appear. I do not know anything special about the Jordan normal form of a companion matrix, perhaps something precise can be said that need not hold for other types of coefficient matrix $B.$
This is half a semester of work if you have already had linear algebra. Anyway, see 
http://en.wikipedia.org/wiki/Ordinary_differential_equation#Fundamental_systems_for_homogeneous_equations_with_constant_coefficients 
