Following the question I asked here and is:
Let $P(\lambda)=(\lambda-\lambda_{0})^{r}$where $r$ is a positive integer. Prove that the equation $P(\frac{d}{dt})x(t)=0$ has solutions $t^{i}e^{\lambda_{0}t},i=0,1,\ldots,r-1$
I now wish to prove the solutions are linearly independent.
I have two questions regarding this:
I learned to prove such independence with the Wronskian, but I am having trouble calculating it in (I calculated the derivatives of $e^{\lambda_{0}t},te^{\lambda_{0}t}$ but its getting too hard when it is a greater power of $t$ since I am getting longer and longer expressions). How can I calculate the Wronskian ?
If I think of the vector space that is the smooth real valued functions then it seems that this set (if I take the power of $t$ to be as big as I want, but finite) is linearly independent. did I deduce right ?
I would appriciate any help!