Power Series with $a_{n}$ and $a_{k}$ 
If $P(x)= \sum_{n=0}^{\infty} a_{n} x^{n}$. It is known that $P$ satisfies: $$P^{\prime}(x)=2xP(x)$$ $\\$ for all $ x\in \mathbb{R}$ and $$P(0)= 1$$
\begin{split}\end{split} (1) Prove that $a_{2k+1}=0$ for every $ k\in \mathbb{\left \{1,2,3,4,...  \right \}}$

Its true that if two power series have the same coefficients then they are equal?
I can demonstrate the first equation transforming the series to an integral, but i cant prove (1) and (2) because $k$ confuses me, I was thinking that $k$ is for $P'(x)$, just as $n$ is for $P(x)$. Also i need orientation with the proof requested.
Thanks.

If $P(x)$ is of degree $n$, then the lhs is of degree $(n-1)$ and the rhs is of degree $(n+1)$.
Since we compare the coefficients of same power, we  have only odd or only even powers.
Now, use the fact that for power $0$, $P(0)\neq 0$ and you have the answer.
