In this question we consider the linear differential system $\mathcal{S} : X' = AX$ associated with the matrix $A = \begin{pmatrix} 1 & -4 & 0 \\ 1 & -2 & -1 \\ 1 & 1 & 0 \end{pmatrix}$. We call trajectories of $\mathcal{S}$ the arcs of the space $\mathbb{R}^3$ parametrized by the solutions of $\mathcal{S}$. We want to determine the rectilinear trajectories and the planar trajectories of $\mathcal{S}$. IV.F.1) Construct an invertible matrix $P$ and a matrix $T = \begin{pmatrix} \alpha & \beta & 0 \\ -\beta & \alpha & 0 \\ 0 & 0 & \gamma \end{pmatrix}$ with $(\alpha, \beta, \gamma)$ in $(\mathbb{R}^*)^3$ such that $P^{-1}AP = T$ and determine a plane $F$ and a line $G$ stable by the endomorphism of $\mathbb{R}^3$ canonically associated with $A$ and supplementary in $\mathbb{R}^3$. IV.F.2) Determine the unique solution of the Cauchy problem $\mathcal{P}_U : \begin{cases} X' = AX \\ X(0) = U \end{cases}$ when $U$ belongs to $G$. IV.F.3) For all $\sigma = (a, b)$ in $\mathbb{R}^2$, we consider the Cauchy problem $\mathcal{C}_{\sigma} : \begin{cases} x' = -x + 2y \\ y' = -2x - y \\ x(0) = a,\ y(0) = b \end{cases}$ and $\varphi = (x, y)$ in $\mathcal{C}^1(\mathbb{R}, \mathbb{R}^2)$ the unique solution of $\mathcal{C}_{\sigma}$. Specify $x'(0)$ and $y'(0)$; show that $x$ and $y$ are solutions of the same linear homogeneous differential equation of second order with constant coefficients and thus deduce $\varphi$ as a function of $a$ and $b$. IV.F.4) Determine the rectilinear trajectories and the planar trajectories of the differential system $X' = AX$.
In this question we consider the linear differential system $\mathcal{S} : X' = AX$ associated with the matrix $A = \begin{pmatrix} 1 & -4 & 0 \\ 1 & -2 & -1 \\ 1 & 1 & 0 \end{pmatrix}$.
We call trajectories of $\mathcal{S}$ the arcs of the space $\mathbb{R}^3$ parametrized by the solutions of $\mathcal{S}$. We want to determine the rectilinear trajectories and the planar trajectories of $\mathcal{S}$.
\textbf{IV.F.1)} Construct an invertible matrix $P$ and a matrix $T = \begin{pmatrix} \alpha & \beta & 0 \\ -\beta & \alpha & 0 \\ 0 & 0 & \gamma \end{pmatrix}$ with $(\alpha, \beta, \gamma)$ in $(\mathbb{R}^*)^3$ such that $P^{-1}AP = T$ and determine a plane $F$ and a line $G$ stable by the endomorphism of $\mathbb{R}^3$ canonically associated with $A$ and supplementary in $\mathbb{R}^3$.
\textbf{IV.F.2)} Determine the unique solution of the Cauchy problem $\mathcal{P}_U : \begin{cases} X' = AX \\ X(0) = U \end{cases}$ when $U$ belongs to $G$.
\textbf{IV.F.3)} For all $\sigma = (a, b)$ in $\mathbb{R}^2$, we consider the Cauchy problem $\mathcal{C}_{\sigma} : \begin{cases} x' = -x + 2y \\ y' = -2x - y \\ x(0) = a,\ y(0) = b \end{cases}$ and $\varphi = (x, y)$ in $\mathcal{C}^1(\mathbb{R}, \mathbb{R}^2)$ the unique solution of $\mathcal{C}_{\sigma}$.
Specify $x'(0)$ and $y'(0)$; show that $x$ and $y$ are solutions of the same linear homogeneous differential equation of second order with constant coefficients and thus deduce $\varphi$ as a function of $a$ and $b$.
\textbf{IV.F.4)} Determine the rectilinear trajectories and the planar trajectories of the differential system $X' = AX$.