In what follows we restrict to the case $n = 2$ from Part I.
Let $x \in C ^ { 2 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { 2 } \right) \right. \right.$ be a solution of equation (1). Show that there exist four real numbers $c _ { 1 } , c _ { 2 } , \varphi _ { 1 } , \varphi _ { 2 }$ such that: $$\forall t \in \left[ 0 , + \infty \left[ , x ( t ) = \sum _ { i = 1 } ^ { 2 } c _ { i } \cos \left( t \sqrt { \lambda _ { i } } + \varphi _ { i } \right) e _ { i } \right. \right.$$ (We recall that the two vectors $e _ { 1 } , e _ { 2 }$ are introduced in Question 4).

We study the differential equation with unknown $y$
$$x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( x ^ { 2 } - n ^ { 2 } \right) y = 0 \tag{III.1}$$
We seek solutions in the set $E = \mathcal { C } ^ { 2 } ( ] 0 , + \infty [ )$.
Let $y$ be a solution in $E$ of (III.1). We set $z ( x ) = \sqrt { x } y ( x )$ for all $x \in ] 0 , + \infty [$.
Show that $z$ is a solution in $E$ of a differential equation of the type
$$z ^ { \prime \prime } + q z = 0 \tag{III.2}$$
with $q \in E$. Specify the expression of the function $q$ and verify that $\lim _ { x \rightarrow + \infty } q ( x ) = 1$.

For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$. Prove that $F$ satisfies on $\mathbb{R}$ a differential equation of the form $F^{\prime} + AF = 0$, where $A$ is a function to be specified.

For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$, and $F$ satisfies $F^{\prime} + AF = 0$ on $\mathbb{R}$. Deduce an expression for $F(x)$.
You may start by differentiating the function $x \mapsto -\frac{1}{8} \ln\left(1 + x^{2}\right) + \frac{\mathrm{i}}{4} \arctan x$.

Let two real numbers $a$ and $c$ such that $c \in D$. Determine the solutions expandable as power series of the differential equation $$x y''(x) + (c - x) y'(x) - a y(x) = 0.$$ We will express these solutions using the Pochhammer symbol and specify the algebraic structure of their set.

For $p \in \mathbb { R } ^ { * }$ we denote by $(E_p)$ the differential equation on $\mathbb { R } _ { + } ^ { * }$ $$\left( E _ { p } \right) : x \left( y ^ { \prime \prime } - y ^ { \prime } \right) + p y = 0 .$$ Let $p \in \mathbb { R } ^ { * }$ and $\left( a _ { n } \right) _ { n \in \mathbb { N } }$ be a sequence of real numbers. We assume that the power series $\sum _ { n \geqslant 0 } a _ { n } x ^ { n }$ has infinite radius of convergence. Show that the function $f : x \mapsto \sum _ { n = 0 } ^ { + \infty } a _ { n } x ^ { n }$ is a solution of $\left( E _ { p } \right)$ if and only if $$\left\{ \begin{array} { l } a _ { 0 } = 0 \\ n ( n + 1 ) a _ { n + 1 } = ( n - p ) a _ { n } , \quad \forall n \in \mathbb { N } ^ { * } \end{array} \right.$$

For $p \in \mathbb { R } ^ { * }$ we denote by $(E_p)$ the differential equation on $\mathbb { R } _ { + } ^ { * }$ $$\left( E _ { p } \right) : x \left( y ^ { \prime \prime } - y ^ { \prime } \right) + p y = 0 .$$ The coefficients of a power series solution satisfy $a_0 = 0$ and $n(n+1)a_{n+1} = (n-p)a_n$ for all $n \in \mathbb{N}^*$. Show that $(E_p)$ has non-identically zero polynomial solutions if and only if $p \in \mathbb { N } ^ { * }$. Show that then, the non-zero polynomial solutions of $(E_p)$ are of degree $p$ and belong to the vector space $E$.

For $p \in \mathbb { R } ^ { * }$ we denote by $(E_p)$ the differential equation on $\mathbb { R } _ { + } ^ { * }$ $$\left( E _ { p } \right) : x \left( y ^ { \prime \prime } - y ^ { \prime } \right) + p y = 0 .$$ We fix a non-zero real $p$ and assume that $p \notin \mathbb { N } ^ { * }$. The coefficients of a power series solution satisfy $a_0 = 0$ and $n(n+1)a_{n+1} = (n-p)a_n$ for all $n \in \mathbb{N}^*$. Justify the existence of sequences $\left( a _ { n } \right) _ { n \in \mathbb { N } } \in \mathbb { R } ^ { \mathbb { N } }$ not identically zero such that the power series $\sum _ { n \geqslant 0 } a _ { n } x ^ { n }$ has infinite radius of convergence and such that the function $x \mapsto \sum _ { n = 0 } ^ { + \infty } a _ { n } x ^ { n }$ is a solution of $(E_p)$.

Problem 1: calculation of an integral
For $x \geqslant 0$ we define $$f ( x ) = \int _ { 0 } ^ { \infty } \frac { e ^ { - t x } } { 1 + t ^ { 2 } } \mathrm {~d} t \quad \text { and } \quad g ( x ) = \int _ { 0 } ^ { \infty } \frac { \sin t } { t + x } \mathrm {~d} t$$
Study of $g$. a. Show that $g ( x )$ is well defined for all $x \geqslant 0$. b. Show that $g$ is of class $C ^ { 2 }$ on $] 0 , \infty [$.
For this you may use the change of variable $u = t + x$ and express $g$ in terms of the functions $C : x \mapsto \int _ { x } ^ { \infty } \frac { \cos u } { u } \mathrm {~d} u$ and $S : x \mapsto \int _ { x } ^ { \infty } \frac { \sin u } { u } \mathrm {~d} u$. c. Determine a linear second-order differential equation satisfied by $f$.

I. Answer the following questions about the differential equation:
$$\cos x \frac { d ^ { 2 } y } { d x ^ { 2 } } - \sin x \frac { d y } { d x } - \frac { y } { \cos x } = 0 \quad \left( - \frac { \pi } { 2 } < x < \frac { \pi } { 2 } \right) .$$

1. A particular solution of Eq. (1) is of the form of $y = ( \cos x ) ^ { m }$ (m is a constant). Find the constant $m$.
2. Find the general solution of Eq. (1), using the solution of Question I.1.

II. Find the value of the following integral:
$$I = \int _ { 1 } ^ { \infty } x ^ { 5 } e ^ { - x ^ { 4 } + 2 x ^ { 2 } - 1 } d x$$
Note that, for a positive constant $\alpha$, the relation $\int _ { 0 } ^ { \infty } e ^ { - \alpha x ^ { 2 } } d x = \frac { 1 } { 2 } \sqrt { \frac { \pi } { \alpha } }$ holds.
III. Express the general solution of the following differential equation in the form of $f ( x , y ) = C$ ( $C$ is a constant) using an appropriate function $f ( x , y )$ :
$$\left( x ^ { 3 } y ^ { n } + x \right) \frac { d y } { d x } + 2 y = 0 \quad ( x > 0 , y > 0 )$$
where $n$ is an arbitrary real constant.