We study the differential equation $$y(x) y'(x) = -4x \tag{E}$$
Show that if $f$ is a solution of $(E)$ on an interval $J$, and if $a$ is a nonzero real number, then the function $h$ defined by $h(x) = a f\left(\frac{x}{a}\right)$ is also a solution of $(E)$ on an interval that one will specify.

We study the differential equation $$y(x) y'(x) = -4x \tag{E}$$
We denote $\mathscr{C}$ the image in $\mathbb{R}^2$ of the application $$\gamma : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}^2, t \mapsto (\cos t, 2\sin t)$$
We denote $g$ the function of a real variable with real values whose graph is $\gamma\left(\left[\frac{\pi}{4}, \pi\right]\right)$.
II.B.1) Determine the domain of definition $\Delta$ of $g$, as well as an expression for $g$. II.B.2) Verify that the restriction of $g$ to the largest open interval contained in $\Delta$ is a solution of $(E)$. II.B.3) Is this a maximal solution? If not, determine a maximal solution $m$ whose graph includes that of $g$.

We study the differential equation $$y(x) y'(x) = -4x \tag{E}$$
II.C.1) Recall the statement of the existence and uniqueness theorem for maximal solutions of a nonlinear scalar differential equation subject to Cauchy conditions. II.C.2) Explain how, and possibly to what extent, this theorem applies to $(E)$. II.C.3) Are the maximal solutions given by this theorem maximal solutions of $(E)$? II.C.4) Deduce from the previous questions the maximal solutions of $(E)$.

For $n \in \mathbb { Z }$, we denote by $\mathcal { E } _ { n }$ the space of functions $f$ in $\mathcal { C } ^ { 2 } \left( \mathbb { R } _ { + } ^ { * } , \mathbb { C } \right)$ such that $$\forall t \in \mathbb { R } _ { + } ^ { * } , \quad t ^ { 2 } f ^ { \prime \prime } ( t ) + t f ^ { \prime } ( t ) - n ^ { 2 } f ( t ) = 0$$
Determine $\mathcal { E } _ { n }$ for $n \in \mathbb { Z }$. We will discuss separately the case $n = 0$.

Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We define $\widetilde { f } ( r , \theta ) = f ( r \cos \theta , r \sin \theta )$ and $\widetilde { g } ( r , \theta ) = g ( r \cos \theta , r \sin \theta )$ on $\mathbb { R } _ { + } ^ { * } \times \mathbb { R }$. For $n \in \mathbb { Z }$, let $$\forall r \in \mathbb { R } _ { + } ^ { * } , \quad c _ { n , f } ( r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } \widetilde { f } ( r , \theta ) e ^ { - i n \theta } \mathrm { ~d} \theta, \quad c _ { n , g } ( r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } \widetilde { g } ( r , \theta ) e ^ { - i n \theta } \mathrm { ~d} \theta$$
Show that $c _ { n , f }$ is of class $\mathcal { C } ^ { 1 }$ on $\mathbb { R } _ { + } ^ { * }$ and satisfies $$\forall r \in \mathbb { R } _ { + } ^ { * } , \quad \left( c _ { n , f } \right) ^ { \prime } ( r ) = \frac { i n } { r } c _ { n , g } ( r )$$

Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We define $\widetilde { f } ( r , \theta ) = f ( r \cos \theta , r \sin \theta )$ and $\widetilde { g } ( r , \theta ) = g ( r \cos \theta , r \sin \theta )$ on $\mathbb { R } _ { + } ^ { * } \times \mathbb { R }$. For $n \in \mathbb { Z }$, let $$\forall r \in \mathbb { R } _ { + } ^ { * } , \quad c _ { n , f } ( r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } \widetilde { f } ( r , \theta ) e ^ { - i n \theta } \mathrm { ~d} \theta, \quad c _ { n , g } ( r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } \widetilde { g } ( r , \theta ) e ^ { - i n \theta } \mathrm { ~d} \theta$$ For $n \in \mathbb { Z }$, $\mathcal { E } _ { n }$ denotes the space of functions $f$ in $\mathcal { C } ^ { 2 } \left( \mathbb { R } _ { + } ^ { * } , \mathbb { C } \right)$ such that $t ^ { 2 } f ^ { \prime \prime } ( t ) + t f ^ { \prime } ( t ) - n ^ { 2 } f ( t ) = 0$ for all $t \in \mathbb { R } _ { + } ^ { * }$.
Show that $c _ { n , f }$ belongs to $\mathcal { E } _ { n }$ and that $c _ { n , f }$ is bounded in a neighbourhood of 0. Deduce the existence of $a _ { n } \in \mathbb { C }$ such that $$\forall r \in \mathbb { R } _ { + } ^ { * } , \quad c _ { n , f } ( r ) = a _ { n } r ^ { | n | }$$

Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We define $\widetilde { f } ( r , \theta ) = f ( r \cos \theta , r \sin \theta )$ and $\widetilde { g } ( r , \theta ) = g ( r \cos \theta , r \sin \theta )$ on $\mathbb { R } _ { + } ^ { * } \times \mathbb { R }$. For $n \in \mathbb { Z }$, let $c _ { n , f } ( r ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } \widetilde { f } ( r , \theta ) e ^ { - i n \theta } \mathrm { ~d} \theta$. We assume that the functions $\frac { \partial f } { \partial x }$ and $\frac { \partial f } { \partial y }$ are bounded on $\mathbb { R } ^ { 2 }$.
If $n \in \mathbb { Z }$, show that the function $\left( c _ { n , f } \right) ^ { \prime }$ is bounded on $\mathbb { R } _ { + } ^ { * }$.

Throughout the problem, $\mathbb { R } ^ { 2 }$ is equipped with the canonical Euclidean inner product denoted $\langle$,$\rangle$ and the associated norm $\| \|$. Let $f$ and $g$ be in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying the Cauchy-Riemann equations: $$\frac { \partial f } { \partial x } = \frac { \partial g } { \partial y } \quad \text { and } \quad \frac { \partial f } { \partial y } = - \frac { \partial g } { \partial x }$$ We assume that the functions $\frac { \partial f } { \partial x }$ and $\frac { \partial f } { \partial y }$ are bounded on $\mathbb { R } ^ { 2 }$.
Show that the functions $\frac { \partial f } { \partial x }$ and $\frac { \partial f } { \partial y }$ are constant.

We denote by $\mathcal { P } _ { 2 }$ the set of polynomial functions of degree $\leqslant 2$ from $\mathbb { R } ^ { 2 }$ to $\mathbb { R }$, that is, the maps from $\mathbb { R } ^ { 2 }$ to $\mathbb { R }$ of the form $$( x , y ) \mapsto a x ^ { 2 } + b x y + c y ^ { 2 } + d x + e y + f \quad \text { where } \quad ( a , b , c , d , e , f ) \in \mathbb { R } ^ { 6 }$$ A function $f \in \mathcal{C}^2(\mathbb{R}^2, \mathbb{R})$ satisfies (1) if and only if $$\forall ( x , y ) \in \mathbb{R}^2, \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$
Determine the functions in $\mathcal { P } _ { 2 }$ satisfying (1) on $\mathbb { R } ^ { 2 }$.

If $I$ is an interval of $\mathbb { R }$, we say that $u \in \mathcal { C } ^ { 1 } ( I , \mathbb { R } )$ satisfies (II.1) on $I$ if and only if $$\forall t \in I , \quad u ( t ) \left( u ( t ) + 2 t u ^ { \prime } ( t ) \right) = - 1$$
By stating precisely the theorem used, show that if $( t _ { 0 } , u _ { 0 } )$ is in $\left( \mathbb { R } ^ { * } \right) ^ { 2 }$, there exist an open interval $I$ of $\mathbb { R }$ containing $t _ { 0 }$ and a function $u \in \mathcal { C } ^ { 1 } ( I , \mathbb { R } )$ such that $u$ is a solution of (II.1) on $I$ and satisfies $u \left( t _ { 0 } \right) = u _ { 0 }$.

If $I$ is an interval of $\mathbb { R }$, we say that $u \in \mathcal { C } ^ { 1 } ( I , \mathbb { R } )$ satisfies (II.1) on $I$ if and only if $$\forall t \in I , \quad u ( t ) \left( u ( t ) + 2 t u ^ { \prime } ( t ) \right) = - 1$$ A function $f \in \mathcal{C}^2(\mathbb{R}^2, \mathbb{R})$ satisfies (1) if and only if $$\forall ( x , y ) \in \mathbb{R}^2, \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$ Let $J$ be a non-empty open interval of $\mathbb { R }$, $\Omega ( J ) = \left\{ ( x , y ) \in \mathbb { R } ^ { 2 } , x y \in J \right\}$, $w$ in $\mathcal { C } ^ { 2 } ( J , \mathbb { R } )$ and $W$ the function defined by $$\forall ( x , y ) \in \Omega ( J ) , \quad W ( x , y ) = w ( x y )$$

1. [II.D.1)] Show that $\Omega ( J )$ is a non-empty open set.
2. [II.D.2)] Show that $W$ is in $\mathcal { C } ^ { 2 } ( \Omega ( J ) , \mathbb { R } )$ and that there is equivalence between
   1. [i.] $W$ satisfies (1) on $\Omega ( J )$,
   2. [ii.] $w ^ { \prime }$ satisfies (II.1) on $J$.
3. [II.D.3)] Show that $W$ is the restriction to $\Omega ( J )$ of a function in $\mathcal { P } _ { 2 }$ if and only if $w$ is affine.

A function $f \in \mathcal{C}^2(\Omega, \mathbb{R})$ satisfies (1) on $\Omega$ if and only if $$\forall ( x , y ) \in \Omega, \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$ Let $\Omega$ be a non-empty open set of $\mathbb { R } ^ { 2 }$, $f$ in $\mathcal { C } ^ { 2 } ( \Omega , \mathbb { R } )$ satisfying (1) on $\Omega$, $( a , b ) \in \mathbb { R } ^ { 2 }$, $\Omega _ { a , b }$ the image of $\Omega$ by the translation of vector $( a , b )$ and $f _ { a , b }$ the function defined on $\Omega _ { a , b }$ by $$\forall ( x , y ) \in \Omega _ { a , b } , \quad f _ { a , b } ( x , y ) = f ( x - a , y - b )$$
Show that $f _ { a , b }$ satisfies (1) on $\Omega _ { a , b }$.

A function $f \in \mathcal{C}^2(\mathbb{R}^2, \mathbb{R})$ satisfies (1) on $\mathbb{R}^2$ if and only if $$\forall ( x , y ) \in \mathbb{R}^2, \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$ We denote by $\mathcal { P } _ { 2 }$ the set of polynomial functions of degree $\leqslant 2$ from $\mathbb { R } ^ { 2 }$ to $\mathbb { R }$.
If $( x _ { 0 } , y _ { 0 } )$ is in $\mathbb { R } ^ { 2 }$, show that there exists an open set $U$ of $\mathbb { R } ^ { 2 }$ containing $( x _ { 0 } , y _ { 0 } )$ such that the set of functions in $\mathcal { C } ^ { 2 } ( U , \mathbb { R } )$ satisfying (1) on $U$ and not coinciding on $U$ with any element of $\mathcal { P } _ { 2 }$ is infinite.

Recall the definition of a $\mathcal { C } ^ { 1 }$-diffeomorphism of $\mathbb { R } ^ { 2 }$ onto $\mathbb { R } ^ { 2 }$ and the theorem characterizing such a diffeomorphism among applications of class $\mathcal { C } ^ { 1 }$ from $\mathbb { R } ^ { 2 }$ to $\mathbb { R } ^ { 2 }$.

We consider $\alpha \in \mathbb { R } _ { + } ^ { * }$ and $F \in \mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } ^ { 2 } \right)$. We assume that for all $( p , h ) \in \mathbb { R } ^ { 2 } \times \mathbb { R } ^ { 2 }$ $$\left\langle d F _ { p } ( h ) , h \right\rangle \geqslant \alpha \| h \| ^ { 2 }$$ Let $p$ and $q$ be in $\mathbb { R } ^ { 2 }$.

1. [III.B.1)] Verify $$F ( q ) - F ( p ) = \int _ { 0 } ^ { 1 } d F _ { p + t ( q - p ) } ( q - p ) \mathrm { d } t$$
2. [III.B.2)] Show $$\langle F ( q ) - F ( p ) , q - p \rangle \geqslant \alpha \| q - p \| ^ { 2 }$$

We consider $\alpha \in \mathbb { R } _ { + } ^ { * }$ and $F \in \mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } ^ { 2 } \right)$. We assume that for all $( p , h ) \in \mathbb { R } ^ { 2 } \times \mathbb { R } ^ { 2 }$ $$\left\langle d F _ { p } ( h ) , h \right\rangle \geqslant \alpha \| h \| ^ { 2 }$$ Let $a \in \mathbb { R } ^ { 2 }$ and $G ^ { a }$ be the map from $\mathbb { R } ^ { 2 }$ to $\mathbb { R }$ defined by $$\forall p \in \mathbb { R } ^ { 2 } , \quad G ^ { a } ( p ) = \| F ( p ) - a \| ^ { 2 }$$

1. [III.C.1)] If $p$ and $h$ are in $\mathbb { R } ^ { 2 }$, compute $d G ^ { a } { } _ { p } ( h )$.
2. [III.C.2)] Show that $G ^ { a } ( p ) \rightarrow + \infty$ when $\| p \| \rightarrow + \infty$.
3. [III.C.3)] Deduce that $G ^ { a }$ attains a global minimum on $\mathbb { R } ^ { 2 }$ at a point $p _ { 0 }$.
4. [III.C.4)] Show that $F \left( p _ { 0 } \right) = a$.

We consider $\alpha \in \mathbb { R } _ { + } ^ { * }$ and $F \in \mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } ^ { 2 } \right)$. We assume that for all $( p , h ) \in \mathbb { R } ^ { 2 } \times \mathbb { R } ^ { 2 }$ $$\left\langle d F _ { p } ( h ) , h \right\rangle \geqslant \alpha \| h \| ^ { 2 }$$
Show that $F$ realizes a $\mathcal { C } ^ { 1 }$-diffeomorphism of $\mathbb { R } ^ { 2 }$ onto $\mathbb { R } ^ { 2 }$.

Let $f$ be in $\mathcal { C } ^ { 2 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying (1) on $\mathbb { R } ^ { 2 }$: $$\forall ( x , y ) \in \mathbb { R } ^ { 2 } , \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$ For $( x , y ) \in \mathbb { R } ^ { 2 }$, let $u ( x , y ) = x + \frac { \partial f } { \partial x } ( x , y ) , v ( x , y ) = y + \frac { \partial f } { \partial y } ( x , y )$ and $F ( x , y ) = ( u ( x , y ) , v ( x , y ) )$. We assume that $\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) > 0$ for all $( x , y ) \in \mathbb { R } ^ { 2 }$.
If $( x , y ) \in \mathbb { R } ^ { 2 }$, show that $\operatorname { Jac } F ( x , y ) - I _ { 2 }$ (where $I _ { 2 }$ denotes the identity matrix of order 2) is symmetric positive semidefinite. Deduce that $F$ is a $\mathcal { C } ^ { 1 }$-diffeomorphism of $\mathbb { R } ^ { 2 }$ onto $\mathbb { R } ^ { 2 }$.

Let $f$ be in $\mathcal { C } ^ { 2 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying (1) on $\mathbb { R } ^ { 2 }$: $$\forall ( x , y ) \in \mathbb { R } ^ { 2 } , \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$ For $( x , y ) \in \mathbb { R } ^ { 2 }$, let $u ( x , y ) = x + \frac { \partial f } { \partial x } ( x , y ) , v ( x , y ) = y + \frac { \partial f } { \partial y } ( x , y )$ and $F ( x , y ) = ( u ( x , y ) , v ( x , y ) )$. We assume that $\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) > 0$ for all $( x , y ) \in \mathbb { R } ^ { 2 }$. Let $r ( x , y ) = \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) , s ( x , y ) = \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y )$ and $t ( x , y ) = \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y )$ so that, for all $( x , y ) \in \mathbb { R } ^ { 2 }$, $r ( x , y ) > 0$ and $r ( x , y ) t ( x , y ) - s ( x , y ) ^ { 2 } = 1$.

1. [IV.B.1)] Show that there exist two functions $\varphi$ and $\psi$ in $\mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ such that $$\forall ( x , y ) \in \mathbb { R } ^ { 2 } , \left\{ \begin{array} { l } \varphi ( u ( x , y ) , v ( x , y ) ) = x - \frac { \partial f } { \partial x } ( x , y ) \\ \psi ( u ( x , y ) , v ( x , y ) ) = - y + \frac { \partial f } { \partial y } ( x , y ) \end{array} \right.$$
2. [IV.B.2)] Compute $\frac { \partial \varphi } { \partial u } ( u ( x , y ) , v ( x , y ) ) , \frac { \partial \varphi } { \partial v } ( u ( x , y ) , v ( x , y ) ) , \frac { \partial \psi } { \partial u } ( u ( x , y ) , v ( x , y ) )$ and $\frac { \partial \psi } { \partial v } ( u ( x , y ) , v ( x , y ) )$ (which we will abbreviate as $\frac { \partial \varphi } { \partial u } , \frac { \partial \varphi } { \partial v } , \frac { \partial \psi } { \partial u }$ and $\frac { \partial \psi } { \partial v }$) in terms of $r ( x , y ) , s ( x , y )$ and $t ( x , y )$ (which we will abbreviate as $r , s$ and $t$).
3. [IV.B.3)] Show that $\frac { \partial \varphi } { \partial u }$ and $\frac { \partial \varphi } { \partial v }$ are bounded on $\mathbb { R } ^ { 2 }$.
4. [IV.B.4)] Show, using the first part, that $\frac { \partial \varphi } { \partial u }$ and $\frac { \partial \varphi } { \partial v }$ are constant.
5. [IV.B.5)] Deduce that $r , s$ and $t$ are constant.

Let $f$ be in $\mathcal { C } ^ { 2 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying (1) on $\mathbb { R } ^ { 2 }$: $$\forall ( x , y ) \in \mathbb { R } ^ { 2 } , \quad \frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } ( x , y ) \times \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } ( x , y ) - \left( \frac { \partial ^ { 2 } f } { \partial x \partial y } ( x , y ) \right) ^ { 2 } = 1$$ We denote by $\mathcal { P } _ { 2 }$ the set of polynomial functions of degree $\leqslant 2$ from $\mathbb { R } ^ { 2 }$ to $\mathbb { R }$.
Show that the only functions in $\mathcal { C } ^ { 2 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } \right)$ satisfying (1) on $\mathbb { R } ^ { 2 }$ belong to $\mathcal { P } _ { 2 }$.

Let $M \in \mathbb{R}_+^* \cup \{+\infty\}$ and $f : {]-\infty, M[} \rightarrow \mathbb{R}$ be a continuous function such that $$\forall (x, y) \in {\left]-\infty, \frac{M}{2}\right[}^2, \quad 2f(x+y) = f(2x) + f(2y) \tag{IV.1}$$
Show that $f'' = 0$, then that the set of continuous solutions of equation (IV.1) forms an $\mathbb{R}$-vector space, for which we will determine a basis.

Let $T$ be a strictly positive real number. We denote by $E(T)$ the set consisting of pairs $(u,v)$ of continuous functions on $[0,T]$ with real values.
A Carnot path controlled by $(u,v) \in E(T)$ is a map $\gamma : [0,T] \rightarrow \mathcal{M}_3(\mathbf{R})$ of class $C^1$ solution of the matrix differential equation: $$\left\{\begin{array}{l} \gamma'(t) = u(t)\gamma(t)M_{1,0,0} + v(t)\gamma(t)M_{0,1,0} \\ \gamma(0) = I_3 \end{array}\right.$$ where $M_{1,0,0}$ and $M_{0,1,0}$ are as defined in the first part.
(a) For all $(u,v) \in E(T)$, justify the existence of a unique Carnot path controlled by $(u,v)$.
(b) Show that $\gamma$ satisfies $$\forall t \in [0,T], \quad \gamma(t) \in \mathbf{H}$$ and explicitly calculate, as a function of $t$, $u$ and $v$, the functions $p(t)$, $q(t)$ and $r(t)$ such that $$\gamma(t) = \exp\left(M_{p(t),q(t),r(t)}\right).$$

For all $(\theta, \varphi) \in \mathbf{R}^2$ and $t \in \mathbf{R}$, we define the controls $$u_{\theta,\varphi}(t) = \sin(\theta - \varphi t) \quad \text{and} \quad v_{\theta,\varphi}(t) = \cos(\theta - \varphi t)$$ and we denote $\gamma_{\theta,\varphi}(t) = \exp\left(M_{p(t),q(t),r(t)}\right)$ the Carnot path controlled by $(u_{\theta,\varphi}, v_{\theta,\varphi})$.
(a) We assume $\varphi \neq 0$. Calculate $p(t)$ and $q(t)$ and verify that $$r(t) = \frac{t\varphi - \sin(t\varphi)}{2\varphi^2}$$
(b) Similarly calculate $\gamma_{\theta,0}(t)$.

The Carnot sphere is the set: $$B(1) = \left\{(p,q,r) \in \mathbf{R}^3 \mid \exists (\theta,\varphi) \in [-\pi,\pi] \times [-2\pi,2\pi], \quad \gamma_{\theta,\varphi}(1) = \exp\left(M_{p,q,r}\right)\right\}.$$
We define the functions $f$ and $g$ on $]0, 2\pi]$ by: $$f(s) = \frac{2(1-\cos s)}{s^2} \quad \text{and} \quad g(s) = \frac{s - \sin s}{2s^2}$$
Show that $f$ and $g$ extend by continuity to $[0, 2\pi]$; that $f$ is then a continuous bijection from $[0, 2\pi]$ onto a set to be specified; and that $g$ attains its maximum at $\pi$.

The Carnot sphere is the set: $$B(1) = \left\{(p,q,r) \in \mathbf{R}^3 \mid \exists (\theta,\varphi) \in [-\pi,\pi] \times [-2\pi,2\pi], \quad \gamma_{\theta,\varphi}(1) = \exp\left(M_{p,q,r}\right)\right\}.$$
The functions $f$ and $g$ on $[0, 2\pi]$ are defined by: $$f(s) = \frac{2(1-\cos s)}{s^2} \quad \text{and} \quad g(s) = \frac{s - \sin s}{2s^2}$$ (extended by continuity at $0$).
Show that if $(p,q,r) \in B(1)$ with $r \geq 0$ then $r = g \circ f^{-1}(p^2 + q^2)$.
State and establish a converse.
One may give the shape of the function $s \mapsto g \circ f^{-1}(s^2)$ for $s \in [0,1]$ and in particular the tangent lines at $s=0$ and $s=1$.