I'm having trouble answering these questions:
Consider the map $D : P_\infty\to P_\infty$ defined by: $D(f)=(x+2)f'$
a) For any non-negative integer k, show that k is an eigenvalue of D by finding a nonzero eigenvector corresponding to k.
I've tried something along the lines of $D(x^k)=(x+2)(kx^{k-1})=kx^k+2kx^{k-1}$, as well as trying $D(a_1+a_2x+a_3x^2+...+a_nx^{n-1})=(x+2)(a_2+a_3x+...+a_n(n-1)x^{n-2})$ but I don't know what to do from here (or if either of these are the right approach at all).
b) Find the kernel of D and state its dimension.
With this I think that that $Ker(D)=\{a_1,a_2,a_3,...,a_n:a_1\in R\Bbb\ \text{and}\ a_2,a_3,...,a_n=0\}$ but I'm not sure how to find out the dimension of this.
Any help would be greatly appreciated.
For part $(a)$, you want a function $f \in P_\infty$ that satisfies $$ D(f) = kf \quad \iff \quad (x+2)f'= kf, $$ which is a simple separable differential equation. If $x=-2$, we see from the above that we must have $f(-2) = 0$, and otherwise if $x\not=-2$, we have $$ \int \frac{\mathrm df}{f} = k\int \frac{\mathrm dx}{x+2} \quad \iff \quad \log f = k \log (x+2) + C \quad \iff \quad f = (x+2)^k, $$ where in the last step we set $C=0$ due to the condition $f(-2)=0$.
For part $(b)$, you want all functions $f \in P_\infty$ such that $$ D(f) = (x+2)f' = 0. $$ Since this should hold for all $x$, we see that $f' = 0$, i.e. $f = \text{const.}$ The dimension of this kernel is evidently equal to $1$.