Consider the differential equation $\frac { d y } { d x } = 3 x + 2 y + 1$. (a) Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$. (b) Find the values of the constants $m , b$, and $r$ for which $y = m x + b + e ^ { r x }$ is a solution to the differential equation. (c) Let $y = f ( x )$ be a particular solution to the differential equation with the initial condition $f ( 0 ) = - 2$. Use Euler's method, starting at $x = 0$ with a step size of $\frac { 1 } { 2 }$, to approximate $f ( 1 )$. Show the work that leads to your answer. (d) Let $y = g ( x )$ be another solution to the differential equation with the initial condition $g ( 0 ) = k$, where $k$ is a constant. Euler's method, starting at $x = 0$ with a step size of 1 , gives the approximation $g ( 1 ) \approx 0$. Find the value of $k$.

A function $f ( x )$ defined for $x > a$ and a quartic function $g ( x )$ with leading coefficient $-1$ satisfy the following conditions. (Here, $a$ is a constant.)
(a) For all real numbers $x > a$, $$( x - a ) f ( x ) = g ( x ).$$ (b) For two distinct real numbers $\alpha , \beta$, the function $f ( x )$ has the same local maximum value $M$ at $x = \alpha$ and $x = \beta$. (Here, $M > 0$)
(c) The number of values of $x$ where the function $f ( x )$ has a local maximum or minimum is greater than the number of values of $x$ where the function $g ( x )$ has a local maximum or minimum. When $\beta - \alpha = 6 \sqrt { 3 }$, find the minimum value of $M$. [4 points]

We consider the differential equation: $$\forall t \in \left[ 0 , + \infty \left[ , A x ^ { \prime \prime } ( t ) = - K x ( t ) \right. \right. \tag{1}$$ with unknown function $x \in C ^ { 2 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { n } \right) \right. \right.$.
Show that $x \in C ^ { 2 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { n } \right) \right. \right.$ is a solution of the differential equation (1) if and only if there exist $2 n$ real numbers $\left( a _ { i } \right) _ { 1 \leq i \leq n } , \left( b _ { i } \right) _ { 1 \leq i \leq n }$ such that: $$\forall t \in \left[ 0 , + \infty \left[ , x ( t ) = \sum _ { i = 1 } ^ { n } \left( a _ { i } \cos \left( t \sqrt { \lambda _ { i } } \right) + b _ { i } \sin \left( t \sqrt { \lambda _ { i } } \right) \right) e _ { i } \right. \right.$$ Deduce that the set of solutions of (1) is a finite-dimensional vector space and specify its dimension.

Let $x , y \in C ^ { 1 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { n } \right) \right. \right.$. Prove that $$\forall t \in \left[ 0 , + \infty \left[ , \frac { d } { d t } ( \langle A x ; y \rangle ) ( t ) = \left\langle A x ^ { \prime } ( t ) ; y ( t ) \right\rangle + \left\langle A x ( t ) ; y ^ { \prime } ( t ) \right\rangle \right. \right.$$

Let $x \in C ^ { 2 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { n } \right) \right. \right.$ be a solution of the differential equation (1). For each real $t \geq 0$ we set, $T \left( x ^ { \prime } \right) ( t ) = \frac { 1 } { 2 } \left\langle A x ^ { \prime } ( t ) ; x ^ { \prime } ( t ) \right\rangle$ and $U ( x ) ( t ) = \frac { 1 } { 2 } \langle K x ( t ) ; x ( t ) \rangle$. Show then that the quantity $T \left( x ^ { \prime } \right) ( t ) + U ( x ) ( t )$ does not depend on $t \in [ 0 , + \infty [$.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{t}{e^t - 1} - 1 + \dfrac{t}{2}$ for all $t \in \mathbb{R}^{+*}$ (extended by continuity at 0).
Determine $E$.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{t}{e^t - 1} - 1 + \dfrac{t}{2}$ for all $t \in \mathbb{R}^{+*}$ (extended by continuity at 0).
Using a series expansion, show that for all $x > 0$, we have $$Lf(x) = \frac{1}{2x^2} - \frac{1}{x} + \sum_{n=1}^{+\infty} \frac{1}{(n+x)^2}.$$

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{t}{e^t - 1} - 1 + \dfrac{t}{2}$ for all $t \in \mathbb{R}^{+*}$ (extended by continuity at 0).
Does $Lf(x) - \dfrac{1}{2x^2} + \dfrac{1}{x}$ admit a finite limit at $0^+$?

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$.
Show that if $E$ is not empty and if $\alpha$ is its infimum (we agree that $\alpha = -\infty$ if $E = \mathbb{R}$), then $Lf$ is of class $C^{\infty}$ on $]\alpha, +\infty[$ and express its successive derivatives using an integral.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$.
In the particular case where $f(t) = e^{-at}t^n$ for all $t \in \mathbb{R}^+$, with $n \in \mathbb{N}$ and $a \in \mathbb{R}$, make explicit $E$, $E^{\prime}$ and calculate $Lf(x)$ for $x \in E^{\prime}$.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$. We assume that $E$ is not empty and that $f$ admits near 0 the following limited expansion of order $n \in \mathbb{N}$: $$f(t) = \sum_{k=0}^{n} \frac{a_k}{k!}t^k + O\left(t^{n+1}\right).$$
IV.C.1) Show that for all $\beta > 0$, we have, as $x$ tends to $+\infty$, the following asymptotic expansion: $$\int_0^{\beta} \left(f(t) - \sum_{k=0}^{n} \frac{a_k}{k!}t^k\right)e^{-tx}\,dt = O\left(x^{-n-2}\right).$$
IV.C.2) Deduce from this that as $x$ tends to infinity, we have the asymptotic expansion: $$Lf(x) = \sum_{k=0}^{n} \frac{a_k}{x^{k+1}} + O\left(x^{-n-2}\right).$$

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 $E = \mathcal { C } ^ { 2 } ( ] 0 , + \infty [ )$.
Using the power series development of $\varphi _ { n }$ (II.C.3), show that $\varphi _ { n }$ is a solution on $[ 0 , + \infty [$ of (III.1).

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 $\lim _ { x \rightarrow + \infty } q ( x ) = 1$.

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 $z$ be a solution of $z^{\prime\prime} + qz = 0$ (III.2).
Justify that if $z$ is a non-zero solution of (III.2), then for $x > 0 , \left( z ( x ) , z ^ { \prime } ( x ) \right) \neq ( 0,0 )$.
Deduce that if $\alpha$ is a zero of $z$, then there exists a strictly positive real $\eta$ such that $\alpha$ is the only point where $z$ vanishes on $I = ] \alpha - \eta , \alpha + \eta [$. In this case, we say that $\alpha$ is an isolated zero of $z$.

Verify that the zeros of $\varphi _ { n }$ on $] 0 , + \infty [$ are isolated.

We study the asymptotic behavior near $+ \infty$ of a solution $z \in E$ of the differential equation defined on $] 0 , + \infty [$, with $\lambda \in \mathbb { R } ^ { * }$ :
$$z ^ { \prime \prime } + \left( 1 + \frac { \lambda } { x ^ { 2 } } \right) z = 0 \tag{III.3}$$
Let $x _ { 0 }$ in $] 0 , + \infty [$.
By considering the differential equation (III.3) in the form $z ^ { \prime \prime } + z = g$ with $g ( x ) = \frac { - \lambda } { x ^ { 2 } } z ( x )$, solve it on $] 0 , + \infty [$ using the method of variation of constants. Deduce that there exist two real numbers $A$ and $B$ such that
$$\forall x \in ] 0 , + \infty \left[ \quad z ( x ) = A \cos ( x ) + B \sin ( x ) + \lambda \int _ { x _ { 0 } } ^ { x } z ( u ) \sin ( u - x ) \frac { \mathrm { d } u } { u ^ { 2 } } \right.$$

We study the asymptotic behavior near $+ \infty$ of a solution $z \in E$ of the differential equation defined on $] 0 , + \infty [$, with $\lambda \in \mathbb { R } ^ { * }$ :
$$z ^ { \prime \prime } + \left( 1 + \frac { \lambda } { x ^ { 2 } } \right) z = 0 \tag{III.3}$$
Let $x _ { 0 }$ in $] 0 , + \infty [$. We set for $x > 0$
$$h ( x ) = \int _ { x _ { 0 } } ^ { x } | z ( u ) | \frac { \mathrm { d } u } { u ^ { 2 } }$$
a) Show that there exist real constants $\mu$ and $M$ such that $h$ satisfies the differential inequality for $x \geqslant x _ { 0 }$
$$h ^ { \prime } ( x ) - \frac { \mu } { x ^ { 2 } } h ( x ) \leqslant \frac { M } { x ^ { 2 } }$$
Specify the constants $\mu$ and $M$ in terms of $A , B$ and $\lambda$.
b) Deduce that $h$ is bounded on $\left[ x _ { 0 } , + \infty [ \right.$ and then that $z$ is bounded on the same interval.
Multiply by $e ^ { \mu / x }$ and integrate the inequality from the previous question.

We study the asymptotic behavior near $+ \infty$ of a solution $z \in E$ of the differential equation defined on $] 0 , + \infty [$, with $\lambda \in \mathbb { R } ^ { * }$ :
$$z ^ { \prime \prime } + \left( 1 + \frac { \lambda } { x ^ { 2 } } \right) z = 0 \tag{III.3}$$
Justify that
$$\int _ { x } ^ { + \infty } z ( u ) \sin ( u - x ) \frac { \mathrm { d } u } { u ^ { 2 } } = O \left( \frac { 1 } { x } \right)$$
near $+ \infty$. Deduce the existence of constants $\alpha$ and $\beta$ such that near $+ \infty$,
$$z ( x ) = \alpha \cos ( x - \beta ) + O \left( \frac { 1 } { x } \right)$$

Let $n \in \mathbb { N }$. Show that there exists a pair of real numbers $( \alpha _ { n } , \beta _ { n } )$ such that for $x \rightarrow + \infty$,
$$\varphi _ { n } ( x ) = \frac { \alpha _ { n } } { \sqrt { x } } \cos \left( x - \beta _ { n } \right) + O \left( \frac { 1 } { x \sqrt { x } } \right)$$

We introduce the differential equation
$$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$
Using the bound from question II.E.2, show that $\varphi _ { 0 } ( 3 ) < 0$. Deduce that $\varphi _ { 0 }$ has a zero $\left. \alpha _ { 0 } \in \right] 0,3 [$.
We admit that this is the first zero of $\varphi _ { 0 }$, that is, $\varphi _ { 0 }$ does not vanish on $] 0 , \alpha _ { 0 } [$.

We introduce the differential equation
$$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$
Using question II.D, show by induction that for all integer $n \geqslant 1$ the function $\varphi _ { n }$ is strictly positive on $] 0 , \alpha _ { 0 } [$.

We introduce the differential equation
$$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$
In this question, we fix $n \in \mathbb { N }$ and $c \in ] 0,1 [$. We set $z ( x ) = \sqrt { x } \varphi _ { n } ( x )$, for $x > 0$.
Let $a > A$. We set for $x > 0 , z _ { 1 } ( x ) = \sin ( c ( x - a ) )$, solution of (IV.1). We define the function $W = z z _ { 1 } ^ { \prime } - z _ { 1 } z ^ { \prime }$.
Verify that for $x > 0 , W ^ { \prime } ( x ) = \left( q ( x ) - c ^ { 2 } \right) z ( x ) z _ { 1 } ( x )$.

We introduce the differential equation
$$z _ { 1 } ^ { \prime \prime } ( x ) + c ^ { 2 } z _ { 1 } ( x ) = 0 \quad \text { with } \quad c > 0 \tag{IV.1}$$
In this question, we fix $n \in \mathbb { N }$ and $c \in ] 0,1 [$. We set $z ( x ) = \sqrt { x } \varphi _ { n } ( x )$, for $x > 0$. Let $a > A$, $z _ { 1 } ( x ) = \sin ( c ( x - a ) )$, and $W = z z _ { 1 } ^ { \prime } - z _ { 1 } z ^ { \prime }$.
We denote $\left. I _ { a } = \right] a , a + \pi / c \left[ \right.$ and assume that $\varphi _ { n }$ has no zeros on $I _ { a }$.
Determine the signs of $W ( a ) , W ( a + \pi / c )$ and of $W ^ { \prime }$ on $I _ { a }$ and reach a contradiction. Deduce that $\varphi _ { n }$ has a zero in every interval $I _ { a }$ with $a > A$.
One may distinguish cases according to the sign of $\varphi _ { n }$ on $I _ { a }$.

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 $n = 0$.

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 $u_a(t)$?