Prove the derivatives of $f(x)$ are linearly dependent I need help with the question below. Please assist!
Suppose $f(x)$ is a funtion such that for some positive integer $n$, $f$ has $n$ linearly dependent derivatives. In other words, if 
$f(x), f'(x), ... , f^{n-1}(x), f^n(x)$ are all linearly dependent functions
then $f(x)$ is expressible in terms of $a, x^k, e^{ax}, \sin(ax), \cos(ax)$, and any combination thereof.
of such functions, where $a$ is a constant and $k$ is a positive integer.
Prove this statement.
 A: Let $n=\min\left\{p\in\mathbb{N}, (f,\ldots,f^{(p)}) \text{ is linearly dependent}\right\}$, there exists $(\alpha_0,\ldots,\alpha_n)\in\mathbb{R}^{n+1}$ such that
$$ \sum_{i=0}^n \alpha_if^{(i)}=0 $$
First $\alpha_n\neq0$ (otherwise this would contradict the definition of $n$), thus if $(a_0,\ldots,a_{n-1})=\left(-\frac{\alpha_0}{\alpha_n},\ldots,-\frac{\alpha_{n-1}}{\alpha_n}\right)$, we have
$$ f^{(n)}=\sum_{i=0}^{n-1}a_if^{(i)} $$
Let $Y=\begin{pmatrix} f\\ \vdots \\ f^{(n-1)} \end{pmatrix}:\mathbb{R}\rightarrow\mathbb{R}^n$ and $$ A=\begin{pmatrix} 0 &1 &0 &\ldots &0 \\
\vdots &\ddots &\ddots &\ddots &\vdots \\
\vdots & &\ddots &\ddots &0 \\
0 &\cdots &\cdots &0 &1 \\
a_0 &\cdots &\cdots &\cdots &a_{n-1} \end{pmatrix} $$
then $Y'=AY$, thus there exists $X\in\mathbb{R}^n$ such that $\forall t\in\mathbb{R},Y(t)=\exp(tA)X$. Let $J$ the Jordan normal form of $A$ :
$$ \begin{pmatrix} J_1 &0 &\cdots &0 \\
0 &\ddots &\ddots &\vdots \\
\vdots &\ddots &\ddots &0 \\
0 &\cdots &0 &J_r
 \end{pmatrix} $$
where
$$ J_i=\begin{pmatrix} \lambda_i &1 &0 &\ldots &0 \\
0 &\ddots &\ddots &\ddots &\vdots \\
\vdots &\ddots &\ddots &\ddots &0 \\
\vdots & &\ddots &\ddots &1 \\
0 &\cdots &\cdots &0 &\lambda_i \end{pmatrix}\in\mathcal{M}_{m_i}(\mathbb{C}) $$
for $\lambda_i\in\mathbb{C}$ and $m_i\in\mathbb{N}^*$. Let $P\in\text{GL}_n(\mathbb{C})$ such that $A=PJP^{-1}$, then $\forall t\in\mathbb{R},Y(t)=P\exp(tJ)P^{-1}X$. However
$$ \exp(tJ)=\begin{pmatrix} \exp(tJ_1) &0 &\cdots &0 \\
0 &\ddots &\ddots &\vdots \\
\vdots &\ddots &\ddots &0 \\
0 &\cdots &0 &\exp(tJ_r)
 \end{pmatrix} $$
and $$ \exp(tJ_i)=e^{\lambda_i t}\begin{pmatrix}
1 &t &\ldots &\ldots &\frac{t^{m_i-1}}{(m_i-1)!} \\
0 &\ddots &\ddots & &\vdots \\
\vdots &\ddots &\ddots &\ddots &\vdots \\
\vdots & &\ddots &\ddots &t \\
0 &\cdots &\cdots &0 &1
\end{pmatrix} $$
After some calculus, you obtain that $f\in\text{Vect}_{\lambda\in\mathbb{C},k\in\mathbb{N}}\left(t\mapsto t^ke^{\lambda t}\right)=\text{Vect}_{\lambda\in\mathbb{R},k\in\mathbb{N}}\left(t\mapsto 1,t\mapsto t^ke^{\lambda t},t\mapsto t^k\cos(\lambda t),t\mapsto t^k\sin(\lambda t)\right)$
