Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system: $$(\mathrm{S}) \quad X' = MX$$ where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$. We assume $n = 2$, we then denote $X = (x; y)$ where $x$ and $y$ are two functions differentiable from $\mathbf{R}$ to $\mathbf{R}$ and we set $z = x + iy$. We assume that there exist real numbers $a$ and $b$ such that $M = \left(\begin{array}{cc} a & b \\ -b & a \end{array}\right)$. Prove that $X$ is a solution of (S) if and only if $z$ is a solution of a first-order linear differential equation to be determined. Deduce an expression, as a function of $t$, of the coordinates of the solutions of (S). Solve the system $X' = BX$ where $B$ is the matrix from question 2).
Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system:
$$(\mathrm{S}) \quad X' = MX$$
where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$.
We assume $n = 2$, we then denote $X = (x; y)$ where $x$ and $y$ are two functions differentiable from $\mathbf{R}$ to $\mathbf{R}$ and we set $z = x + iy$.
We assume that there exist real numbers $a$ and $b$ such that $M = \left(\begin{array}{cc} a & b \\ -b & a \end{array}\right)$.
Prove that $X$ is a solution of (S) if and only if $z$ is a solution of a first-order linear differential equation to be determined. Deduce an expression, as a function of $t$, of the coordinates of the solutions of (S).
Solve the system $X' = BX$ where $B$ is the matrix from question 2).