<s>[INST] ODE system solving by sequence of functions Let $y' = Ay$ where $A = \begin{pmatrix} 0&1 \\ -1& 0 \end{pmatrix}$ and $y( 0 ) = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$. Consider the map $$G: C(\mathbb{R},\mathbb{R}^2) \to C(\mathbb{R},\mathbb{R}^2),  G(\phi)(x) = \begin{pmatrix} 1 \\ 0 \end{pmatrix} + \int_0^x A\phi(t) dt$$.
With $\phi_0(x) = \begin{pmatrix} 1\\0\end{pmatrix}$ and $\phi_{n+1} = G(\phi_n)$, how to find and prove the general formula for $(\phi_n)_{n\in\mathbb{N}}$?
 [/INST] Note that
\begin{equation}
\begin{split}
& \phi_{n+1}' = A\phi_n \\
\implies & \phi_{n}^{(n)} = A^n\phi_0,
\end{split}
\end{equation}
which is a constant vector. This shows $\phi_n$ is a vector of polynomials with degree at most $n$. The $2(n+1)$ coefficients can be found by a  system of linear equations: for $s = 0, 1, \ldots k$,
\begin{equation}
\begin{split}
\phi_n^{(s)} = A^{s}\phi_{n-s} \implies \phi_n^{(s)}(0) = A^s \begin{pmatrix} 1 \\ 0 \end{pmatrix}.
\end{split}
\end{equation} 
For each $s$, the above formula contributes to $2$ linear equations in the coefficients.
More precisely, if we let 
$$p_s(x) = \dfrac{1}{s!}x^s, s = 0, 1, 2, \ldots, n,$$
and write
$$\phi_n(x) = \begin{pmatrix} \sum_{s=0}^n C_s p_s(x) \\ \sum_{s=0}^n K_s p_s(x) \end{pmatrix}.$$
Then the coefficients $C_s, K_s$ are given by
$$ \phi_n^{(s)}(0) = \begin{pmatrix} C_s \\ K_s \end{pmatrix} = A^s \begin{pmatrix} 1 \\ 0 \end{pmatrix}. $$
It is now reduced to find a general formula for $A^s \begin{pmatrix} 1 \\ 0 \end{pmatrix}$.
</s>