We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$ In other words, $u_a$ is the unique function of class $C^1$ from $\mathbb{R}$ to $\mathbb{R}^2$ such that $u_a(0) = a$ and, for every real $t$, $u_a'(t) = A u_a(t)$.
We assume $A$ is diagonal of the form $$A = \operatorname{diag}(\lambda_1, \lambda_2) = \begin{pmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{pmatrix}$$
What is $u_a(t)$?

We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$ In other words, $u_a$ is the unique function of class $C^1$ from $\mathbb{R}$ to $\mathbb{R}^2$ such that $u_a(0) = a$ and, for every real $t$, $u_a'(t) = A u_a(t)$.
We assume $A$ is diagonal of the form $$A = \operatorname{diag}(\lambda_1, \lambda_2) = \begin{pmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{pmatrix}$$
Let $a$ and $b$ be two elements of $\mathbb{R}^2$ and let $t$ be a real number. Show that $$\operatorname{det}\left(u_a(t), u_b(t)\right) = \exp\left(t \operatorname{div}_f(a)\right) \operatorname{det}\left(u_a(0), u_b(0)\right)$$

We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$ In other words, $u_a$ is the unique function of class $C^1$ from $\mathbb{R}$ to $\mathbb{R}^2$ such that $u_a(0) = a$ and, for every real $t$, $u_a'(t) = A u_a(t)$.
We assume $A$ is diagonal of the form $$A = \operatorname{diag}(\lambda_1, \lambda_2) = \begin{pmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{pmatrix}$$
Use the result of II.B.2 to interpret the sign of $\operatorname{div}_f(a)$ in terms of the direction of variation of the area of a certain parallelogram as a function of $t$.

We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$ We still assume that $A = \operatorname{diag}(\lambda_1, \lambda_2)$.
We set $a = (a_1, a_2)$ and $u_a(t) = (x_1(t), x_2(t))$. We assume that $\lambda_1 \neq 0$ and $a_1 > 0$. Determine a function $\theta_a$ such that $x_2(t) = \theta_a(x_1(t))$ for every real $t$.

We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$ We still assume that $A = \operatorname{diag}(\lambda_1, \lambda_2)$.
In this question, $a = (2,1)$ and $b = (1,2)$.
For each of the following cases, illustrate on the same figure the graphs of the functions $\theta_a$, $\theta_b$ and $\theta_{a+b}$, as well as the parallelograms with vertices $(0,0)$, $u_a(t)$, $u_b(t)$ and $u_a(t) + u_b(t)$ for $t = 0$ and a strictly positive value of $t$.
a) $\lambda_1 = 1$ and $\lambda_2 = 2$.
b) $\lambda_1 = 1$ and $\lambda_2 = -2$.
c) $\lambda_1 = 1$ and $\lambda_2 = -1$.

We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$
Revisit questions II.B.1 and II.B.2 in the case where $A$ is triangular of the form $$A = \begin{pmatrix} \lambda & \mu \\ 0 & \lambda \end{pmatrix}$$

We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$
Show that the relation $$\operatorname{det}\left(u_a(t), u_b(t)\right) = \exp\left(t \operatorname{div}_f(a)\right) \operatorname{det}\left(u_a(0), u_b(0)\right)$$ holds when the matrix $A$ has a characteristic polynomial that splits over $\mathbb{R}$.

We denote by $A$ a real square matrix of size 2 and we set, for all $x$ in $\mathbb{R}^2$, $f(x) = Ax$. For $a$ in $\mathbb{R}^2$, we denote by $u_a(t)$ the solution on $\mathbb{R}$ of the Cauchy problem $$X' = AX, \quad X(0) = a$$
Extend the result $$\operatorname{det}\left(u_a(t), u_b(t)\right) = \exp\left(t \operatorname{div}_f(a)\right) \operatorname{det}\left(u_a(0), u_b(0)\right)$$ to the case of an arbitrary real $2 \times 2$ matrix.

We assume that $m = 0$ and there exists $\lambda > 0$ such that $G(0) = (0,0,2\lambda)$, $G^{\prime}(0) = (1,0,0)$, and $G$ satisfies $$\forall x \in \mathbb{R}, G^{\prime\prime\prime}(x) + \left(\lambda^{2} + \frac{x^{2}}{4}\right) G^{\prime}(x) - \frac{x}{4} G(x) = 0$$
(a) Write the linear differential equation $Y^{\prime\prime\prime} + \left(\lambda^{2} + \frac{x^{2}}{4}\right) Y^{\prime} - \frac{x}{4} Y = 0$, where $Y \in C^{3}(\mathbb{R}, \mathbb{R}^{3})$, in the form of a differential system $X^{\prime} = AX$, where $X \in C^{1}(\mathbb{R}, \mathcal{M}_{n,1}(\mathbb{R}))$ and where $A \in C(\mathbb{R}, \mathcal{M}_{n}(\mathbb{R}))$, with $n \in \mathbb{N}^{*}$. We will specify $n$ and $A$.
(b) Show that the coordinates $G_{1}, G_{2}, G_{3}$ of $G$ satisfy $$\forall x \in \mathbb{R}, G_{1}(-x) = -G_{1}(x),\quad G_{2}(-x) = G_{2}(x),\quad G_{3}(-x) = G_{3}(x)$$
(c) Show that for all $x \in \mathbb{R}$, $\|G(x)\|^{2} = x^{2} + 4\lambda^{2}$.
(d) Establish that if $G_{1}$ does not vanish on $\mathbb{R}^{*}$, then $G$ is an injective application on $\mathbb{R}$.

Solve the differential system $$\left\{ \begin{array}{l} x'(t) = x(t) - \cos(t) y(t) \\ y'(t) = \cos(t) x(t) + y(t) \end{array} \right.$$ and determine its normal form (see question 16d).

Solve the differential system $$\left\{\begin{array}{l} x'(t) = x(t) - \cos(t) y(t) \\ y'(t) = \cos(t) x(t) + y(t) \end{array}\right.$$ and determine its normal form (see question 16d).