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.
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 $f\in\text{Vect}_{\lambda\in\mathbb{C},k\in\mathbb{N}}\left(t\mapsto t^ke^{\lambda t}\right)=\text{Vect}_{\lambda\in\mathbb{R},k\in\mathbb{N}}\left(t\mapsto 1,t\mapsto t^ke^{\lambda t},t\mapsto t^k\cos(\lambda t),t\mapsto t^k\sin(\lambda t)\right)$