grandes-ecoles

Papers (191)
2025
centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38
2024
centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26
2023
centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15
2022
centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23
2021
centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44
2020
centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18
2019
centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26
2018
centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24
2017
centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26
2016
centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20
2015
centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24
2013
centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18
2011
centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10
2010
centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27
2022 centrale-maths2__psi

45 maths questions

Q1 Proof Proof of Set Membership, Containment, or Structural Property View
Show that, for all $\alpha \in \mathbb { R } _ { + } ^ { * } , p _ { \alpha }$ belongs to $E$, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges, and $p_\alpha$ is the function $t \mapsto t^\alpha$.
Q2 Proof Proof of Equivalence or Logical Relationship Between Conditions View
Let $P$ be a polynomial function not identically zero with real coefficients. Show that the restriction of $P$ to $\mathbb { R } _ { + } ^ { * }$ belongs to $E$ if and only if $P ( 0 ) = 0$, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges.
Q3 Proof Proof of Equivalence or Logical Relationship Between Conditions View
Let $a$ and $b$ be two real numbers. Show that the function $\left\lvert\, \begin{array} { r l l } \mathbb { R } _ { + } ^ { * } & \rightarrow & \mathbb { R } \\ t & \mapsto & a \mathrm { e } ^ { t } + b \end{array} \right.$ belongs to $E$ if and only if $a = b = 0$, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges.
Q5 Proof Proof of Set Membership, Containment, or Structural Property View
For all $x \in \mathbb { R } _ { + } ^ { * }$ and all $t \in \mathbb { R } _ { + } ^ { * }$, we denote $k _ { x } ( t ) = \mathrm { e } ^ { \min ( x , t ) } - 1$ where $\min ( x , t )$ denotes the smaller of the real numbers $x$ and $t$. Draw a graph of the function $k _ { x }$. Show that $k _ { x }$ belongs to $E$, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges.
Q6 Proof Deduction or Consequence from Prior Results View
We assume that $f$ is a function from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ of class $\mathcal { C } ^ { 1 }$ satisfying $$\left\{ \begin{array} { l } \lim _ { x \rightarrow 0 } f ( x ) = 0 \\ \exists C > 0 ; \forall x > 0 , \quad \left| f ^ { \prime } ( x ) \right| \leqslant C \frac { \mathrm { e } ^ { x / 2 } } { \sqrt { x } } \end{array} \right.$$ For $x \in \mathbb { R } _ { + } ^ { * }$, we set $\Phi ( x ) = \frac { 4 \sqrt { x } \mathrm { e } ^ { x / 2 } } { 1 + x } - \int _ { 0 } ^ { x } \frac { \mathrm { e } ^ { t / 2 } } { \sqrt { t } } \mathrm {~d} t$. Show that $\Phi$ is of class $\mathcal { C } ^ { 1 }$ on $\mathbb { R } _ { + } ^ { * }$, that $\lim _ { x \rightarrow 0 } \Phi ( x ) = 0$ and that, for all $x > 0 , \Phi ^ { \prime } ( x ) \geqslant 0$. Deduce that $\Phi ( x ) \geqslant 0$ for all $x > 0$.
Q7 Differential equations Qualitative Analysis of DE Solutions View
We assume that $f$ is a function from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ of class $\mathcal { C } ^ { 1 }$ satisfying $$\left\{ \begin{array} { l } \lim _ { x \rightarrow 0 } f ( x ) = 0 \\ \exists C > 0 ; \forall x > 0 , \quad \left| f ^ { \prime } ( x ) \right| \leqslant C \frac { \mathrm { e } ^ { x / 2 } } { \sqrt { x } } \end{array} \right.$$ Show that, for all $x > 0 , | f ( x ) | \leqslant 4 C \frac { \sqrt { x } \mathrm { e } ^ { x / 2 } } { 1 + x }$.
Q8 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View
We assume that $f$ is a function from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ of class $\mathcal { C } ^ { 1 }$ satisfying $$\left\{ \begin{array} { l } \lim _ { x \rightarrow 0 } f ( x ) = 0 \\ \exists C > 0 ; \forall x > 0 , \quad \left| f ^ { \prime } ( x ) \right| \leqslant C \frac { \mathrm { e } ^ { x / 2 } } { \sqrt { x } } \end{array} \right.$$ Using the result of Question 7, deduce that $f \in E$, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges.
Q9 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View
Show that, if $f$ and $g$ are two functions in $E$, then the integral $\int _ { 0 } ^ { + \infty } f ( t ) g ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ is absolutely convergent, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges.
Q10 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View
Deduce that $E$ is a vector subspace of the vector space $\mathcal { C } \left( \mathbb { R } _ { + } ^ { * } , \mathbb { R } \right)$ of continuous functions on $\mathbb { R } _ { + } ^ { * }$ with values in $\mathbb { R }$, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges.
Q11 Continuous Probability Distributions and Random Variables Verification of Probability Measure or Inner Product Properties View
For all functions $f \in E$ and $g \in E$, we set $\langle f \mid g \rangle = \int _ { 0 } ^ { + \infty } f ( t ) g ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Show that this defines an inner product on $E$.
Q12 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View
The norm $\| \cdot \|$ associated with the inner product $\langle f \mid g \rangle = \int _ { 0 } ^ { + \infty } f ( t ) g ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ is defined for all functions $f \in E$ by $$\| f \| = \left( \int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t \right) ^ { 1 / 2 }$$ Show that $\lim _ { x \rightarrow 0 } \left\| k _ { x } \right\| = 0$. We recall that, for all $x > 0 , k _ { x } ( t ) = \mathrm { e } ^ { \min ( x , t ) } - 1$.
Q13 Reduction Formulae Compute a Base Case or Specific Value of a Parametric Integral View
Show that, for all $k \in \mathbb { N } , \int _ { 0 } ^ { + \infty } t ^ { k } \mathrm { e } ^ { - t } \mathrm {~d} t = k !$.
Q14 Proof True/False Justification View
The functions $p _ { \alpha }$ are defined by $p_\alpha : t \mapsto t^\alpha$ for $\alpha \in \mathbb{R}_+^*$, and the inner product on $E$ is $\langle f \mid g \rangle = \int _ { 0 } ^ { + \infty } f ( t ) g ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Is the family $\left( p _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ an orthogonal family of $E$?
Q15 Proof Computation of a Limit, Value, or Explicit Formula View
To each function $f \in E$, we associate the function $U ( f )$ defined for all $x > 0$ by $$U ( f ) ( x ) = \left\langle k _ { x } \mid f \right\rangle = \int _ { 0 } ^ { + \infty } \left( \mathrm { e } ^ { \min ( x , t ) } - 1 \right) f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$$ Using the Cauchy-Schwarz inequality, show that for all functions $f \in E$, $$\lim _ { \substack { x \rightarrow 0 \\ x > 0 } } U ( f ) ( x ) = 0$$
Q16 Integration by Parts Integral Involving a Parameter or Operator Identity View
To each function $f \in E$, we associate the function $U ( f )$ defined for all $x > 0$ by $$U ( f ) ( x ) = \left\langle k _ { x } \mid f \right\rangle = \int _ { 0 } ^ { + \infty } \left( \mathrm { e } ^ { \min ( x , t ) } - 1 \right) f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$$ Show that for all functions $f \in E$ and for all $x > 0$, $$U ( f ) ( x ) = \int _ { 0 } ^ { x } \left( 1 - \mathrm { e } ^ { - t } \right) \frac { f ( t ) } { t } \mathrm {~d} t + \left( \mathrm { e } ^ { x } - 1 \right) \int _ { x } ^ { + \infty } f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$$
Q17 Differential equations Higher-Order and Special DEs (Proof/Theory) View
To each function $f \in E$, we associate the function $U ( f )$ defined for all $x > 0$ by $$U ( f ) ( x ) = \int _ { 0 } ^ { x } \left( 1 - \mathrm { e } ^ { - t } \right) \frac { f ( t ) } { t } \mathrm {~d} t + \left( \mathrm { e } ^ { x } - 1 \right) \int _ { x } ^ { + \infty } f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$$ Let $f \in E$. Show that $U ( f )$ is of class $\mathcal { C } ^ { 1 }$ on $\mathbb { R } _ { + } ^ { * }$ and satisfies, for all $x > 0$, $$( U ( f ) ) ^ { \prime } ( x ) = \mathrm { e } ^ { x } \int _ { x } ^ { + \infty } f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$$
Q18 Second order differential equations Solving non-homogeneous second-order linear ODE View
To each function $f \in E$, we associate the function $U ( f )$ with derivative $U(f)^\prime(x) = \mathrm { e } ^ { x } \int _ { x } ^ { + \infty } f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Let $f \in E$. Show that $U ( f )$ is of class $\mathcal { C } ^ { 2 }$ on $\mathbb { R } _ { + } ^ { * }$ and that the function $U ( f )$ is a solution on $\mathbb { R } _ { + } ^ { * }$ of the differential equation $$y ^ { \prime \prime } - y ^ { \prime } = - \frac { f ( x ) } { x }$$
Q19 Integration using inverse trig and hyperbolic functions View
To each function $f \in E$, we associate the function $U ( f )$ with derivative $U(f)^\prime(x) = \mathrm { e } ^ { x } \int _ { x } ^ { + \infty } f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Show that for all $f \in E$ and for all $x > 0$, $$\left| U ( f ) ^ { \prime } ( x ) \right| \leqslant \mathrm { e } ^ { x } \| f \| \left( \int _ { x } ^ { + \infty } \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t \right) ^ { 1 / 2 } \leqslant \| f \| \frac { \mathrm { e } ^ { x / 2 } } { \sqrt { x } }$$
Q20 Reduction Formulae Bound or Estimate a Parametric Integral View
To each function $f \in E$, we associate the function $U ( f )$ defined for all $x > 0$ by $U ( f ) ( x ) = \int _ { 0 } ^ { + \infty } \left( \mathrm { e } ^ { \min ( x , t ) } - 1 \right) f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. It has been shown that $\left| U ( f ) ^ { \prime } ( x ) \right| \leqslant \| f \| \frac { \mathrm { e } ^ { x / 2 } } { \sqrt { x } }$ and $\lim_{x\to 0} U(f)(x) = 0$. Deduce from the above that $U$ is an endomorphism of $E$ and that, for all $f \in E$ and all $x > 0$, $$| U ( f ) ( x ) | \leqslant 4 \| f \| \frac { \sqrt { x } \mathrm { e } ^ { x / 2 } } { 1 + x }$$
Q21 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View
To each function $f \in E$, we associate the function $U ( f )$ defined for all $x > 0$ by $U ( f ) ( x ) = \int _ { 0 } ^ { + \infty } \left( \mathrm { e } ^ { \min ( x , t ) } - 1 \right) f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. It has been shown that $| U ( f ) ( x ) | \leqslant 4 \| f \| \frac { \sqrt { x } \mathrm { e } ^ { x / 2 } } { 1 + x }$ for all $x > 0$. Deduce that $$\| U ( f ) \| \leqslant 4 \| f \|.$$
Q22 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View
To each function $f \in E$, we associate the function $U ( f )$ defined for all $x > 0$ by $U ( f ) ( x ) = \int _ { 0 } ^ { + \infty } \left( \mathrm { e } ^ { \min ( x , t ) } - 1 \right) f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Show that $U$ is injective.
Q23 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
To each function $f \in E$, we associate the function $U ( f )$ defined for all $x > 0$ by $U ( f ) ( x ) = \int _ { 0 } ^ { + \infty } \left( \mathrm { e } ^ { \min ( x , t ) } - 1 \right) f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Is the endomorphism $U$ surjective?
Q24 Indefinite & Definite Integrals Antiderivative Verification and Construction View
We fix two functions $f$ and $g$ in $E$. For $x > 0$, we set $$F ( x ) = - U ( f ) ^ { \prime } ( x ) \mathrm { e } ^ { - x }$$ where $U(f)^\prime(x) = \mathrm { e } ^ { x } \int _ { x } ^ { + \infty } f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Verify that $F$ is an antiderivative of $x \mapsto f ( x ) \frac { \mathrm { e } ^ { - x } } { x }$ on the interval $\mathbb { R } _ { + } ^ { * }$.
Q25 Continuous Probability Distributions and Random Variables Expectation and Moment Inequality Proof View
We fix two functions $f$ and $g$ in $E$. For $x > 0$, we set $F ( x ) = - U ( f ) ^ { \prime } ( x ) \mathrm { e } ^ { - x }$. It has been shown that $| U ( g ) ( x ) | \leqslant 4 \| g \| \frac { \sqrt { x } \mathrm { e } ^ { x / 2 } } { 1 + x }$ and $\left| U ( f ) ^ { \prime } ( x ) \right| \leqslant \| f \| \frac { \mathrm { e } ^ { x / 2 } } { \sqrt { x } }$. Show that for all $x > 0 , | F ( x ) U ( g ) ( x ) | \leqslant \frac { 4 \| f \| \| g \| } { 1 + x }$.
Q26 Continuous Probability Distributions and Random Variables Expectation and Moment Inequality Proof View
We fix two functions $f$ and $g$ in $E$. For $x > 0$, we set $F ( x ) = - U ( f ) ^ { \prime } ( x ) \mathrm { e } ^ { - x }$ where $U(f)^\prime(x) = \mathrm { e } ^ { x } \int _ { x } ^ { + \infty } f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Show that for all $x \in ] 0,1] , | F ( x ) | \leqslant \| f \| \left( \mathrm { e } ^ { - 1 } - \ln ( x ) \right) ^ { 1 / 2 }$. One may use Question 19.
Q27 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View
We fix two functions $f$ and $g$ in $E$. For $x > 0$, we set $F ( x ) = - U ( f ) ^ { \prime } ( x ) \mathrm { e } ^ { - x }$. Show the existence and calculate the values of the limits at $0$ and at $+ \infty$ of the function $t \mapsto F ( t ) U ( g ) ( t )$.
Q28 Integration by Parts Inner Product or Orthogonality Proof via Integration by Parts View
We fix two functions $f$ and $g$ in $E$. For $x > 0$, we set $F ( x ) = - U ( f ) ^ { \prime } ( x ) \mathrm { e } ^ { - x }$, which is an antiderivative of $x \mapsto f ( x ) \frac { \mathrm { e } ^ { - x } } { x }$. The limits of $t \mapsto F(t)U(g)(t)$ at $0$ and $+\infty$ are both $0$. Show that $$\langle f \mid U ( g ) \rangle = \int _ { 0 } ^ { + \infty } U ( f ) ^ { \prime } ( t ) U ( g ) ^ { \prime } ( t ) \mathrm { e } ^ { - t } \mathrm {~d} t.$$
Q29 Integration by Parts Inner Product or Orthogonality Proof via Integration by Parts View
We fix two functions $f$ and $g$ in $E$. Using the result of Question 28, deduce that $\langle f \mid U ( g ) \rangle = \langle U ( f ) \mid g \rangle$.
Q30 Second order differential equations Solving homogeneous second-order linear ODE View
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 $$\left\{ \begin{array} { l } a _ { 0 } = 0 \\ n ( n + 1 ) a _ { n + 1 } = ( n - p ) a _ { n } , \quad \forall n \in \mathbb { N } ^ { * } \end{array} \right.$$
Q31 Second order differential equations Solving homogeneous second-order linear ODE View
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 $E$.
Q32 Second order differential equations Verifying a particular solution satisfies a second-order ODE View
For $p \in \mathbb { N } ^ { * }$, consider a polynomial $P \in \mathbb { R } [ X ]$ such that the polynomial function $x \mapsto P ( x )$ is a solution of the equation $\left( E _ { p } \right) : x \left( y ^ { \prime \prime } - y ^ { \prime } \right) + p y = 0$. For all $x \in \mathbb { R }$, we denote $h ( x ) = \mathrm { e } ^ { - x } P ( x )$. Show that the function $h$ is a solution of the differential equation $x \left( y ^ { \prime \prime } + y ^ { \prime } \right) + p y = 0$ on $\mathbb { R } _ { + } ^ { * }$.
Q33 Taylor series Prove smoothness or power series expandability of a function View
For $p \in \mathbb { N } ^ { * }$, consider a polynomial $P \in \mathbb { R } [ X ]$ such that the polynomial function $x \mapsto P ( x )$ is a solution of the equation $\left( E _ { p } \right) : x \left( y ^ { \prime \prime } - y ^ { \prime } \right) + p y = 0$. For all $x \in \mathbb { R }$, we denote $h ( x ) = \mathrm { e } ^ { - x } P ( x )$. Justify that the function $h$ is developable as a power series on $\mathbb { R }$.
Q34 Sequences and Series Recurrence Relations and Sequence Properties View
For $p \in \mathbb { N } ^ { * }$, let $h(x) = \mathrm{e}^{-x} P(x)$ where $P$ is a polynomial solution of $(E_p)$. We denote by $\left( b _ { n } \right) _ { n \in \mathbb { N } }$ the sequence of coefficients of the power series expansion of $h$, so that for all $x \in \mathbb { R }$, $h ( x ) = \sum _ { n = 0 } ^ { + \infty } b _ { n } x ^ { n }$. These coefficients satisfy $$\left\{ \begin{array} { l } b _ { 0 } = 0 \\ n ( n + 1 ) b _ { n + 1 } = - ( n + p ) b _ { n } , \quad \forall n \in \mathbb { N } ^ { * } . \end{array} \right.$$ Establish that, for all $n \in \mathbb { N } ^ { * } , b _ { n } = \frac { ( - 1 ) ^ { n - 1 } ( n + p - 1 ) ! } { p ! n ! ( n - 1 ) ! } b _ { 1 }$.
Q35 Taylor series Prove smoothness or power series expandability of a function View
For $p \in \mathbb { N } ^ { * }$, let $h(x) = \mathrm{e}^{-x} P(x)$ where $P$ is a polynomial solution of $(E_p)$, with power series coefficients satisfying $b _ { n } = \frac { ( - 1 ) ^ { n - 1 } ( n + p - 1 ) ! } { p ! n ! ( n - 1 ) ! } b _ { 1 }$ for all $n \in \mathbb{N}^*$. We set $g _ { p } ( x ) = x ^ { p - 1 } \mathrm { e } ^ { - x }$. Justify that $g _ { p } ^ { ( p ) }$ is developable as a power series and deduce from Question 34 that, for all $x \in \mathbb { R }$, $$P ( x ) = C x \mathrm { e } ^ { x } g _ { p } ^ { ( p ) } ( x )$$ where $C$ is a real constant whose expression in terms of $b _ { 1 }$ and $p$ we will specify.
Q36 Second order differential equations Solving homogeneous second-order linear ODE View
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 $(E_p)$.
Q37 Sequences and Series Power Series Expansion and Radius of Convergence View
For $p \notin \mathbb{N}^*$, $p \neq 0$, let $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ be a non-zero power series solution of $(E_p) : x(y'' - y') + py = 0$, with coefficients satisfying $n(n+1)a_{n+1} = (n-p)a_n$ for all $n \in \mathbb{N}^*$. Show that there exists a natural integer $q > p$ such that, for all integer $n \geqslant q$, $$\left| a _ { n + 1 } \right| \geqslant \frac { \left| a _ { n } \right| } { 2 ( n + 1 ) }.$$
Q38 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View
For $p \notin \mathbb{N}^*$, $p \neq 0$, let $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ be a non-zero power series solution of $(E_p)$, with $q$ a natural integer such that $q > p$ and $\left| a _ { n + 1 } \right| \geqslant \frac { \left| a _ { n } \right| } { 2 ( n + 1 ) }$ for all $n \geqslant q$. Deduce that, for all integer $n \geqslant q , \left| a _ { n } \right| \geqslant \frac { q ! \left| a _ { q } \right| } { 2 ^ { n - q } n ! }$.
Q39 Sequences and Series Power Series Expansion and Radius of Convergence View
For $p \notin \mathbb{N}^*$, $p \neq 0$, let $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ be a non-zero power series solution of $(E_p)$, with $q$ a natural integer such that $q > p$ and $\left| a _ { n } \right| \geqslant \frac { q ! \left| a _ { q } \right| } { 2 ^ { n - q } n ! }$ for all $n \geqslant q$. Show that the function $\psi : \left\lvert\, \begin{array} { c c c } \mathbb { R } _ { + } ^ { * } & \rightarrow & \mathbb { R } \\ x & \mapsto & \sum _ { n = q } ^ { + \infty } \left| a _ { n } \right| x ^ { n } \end{array} \right.$ is not an element of $E$.
Q40 Sequences and Series Power Series Expansion and Radius of Convergence View
For $p \notin \mathbb{N}^*$, $p \neq 0$, let $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ be a non-zero power series solution of $(E_p)$. Using the result of Question 39, deduce finally that the function $f$ is not an element of $E$.
Q41 Differential equations Eigenvalue Problems and Operator-Based DEs View
To each function $f \in E$, we associate the endomorphism $U$ of $E$ defined for all $x > 0$ by $$U ( f ) ( x ) = \int _ { 0 } ^ { + \infty } \left( \mathrm { e } ^ { \min ( x , t ) } - 1 \right) f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$$ Is the real number $0$ an eigenvalue of $U$?
Q42 Differential equations Higher-Order and Special DEs (Proof/Theory) View
To each function $f \in E$, we associate the endomorphism $U$ of $E$ defined for all $x > 0$ by $$U ( f ) ( x ) = \int _ { 0 } ^ { + \infty } \left( \mathrm { e } ^ { \min ( x , t ) } - 1 \right) f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$$ It has been shown that $U(f)$ satisfies $y'' - y' = -f(x)/x$ on $\mathbb{R}_+^*$. Let $\lambda \in \mathbb { R } ^ { * }$. We assume that $\lambda$ is an eigenvalue of $U$. Let $f$ be an eigenvector associated with it. Show that $f$ is a solution of the differential equation $(E_{1/\lambda}) : x(y'' - y') + \frac{1}{\lambda} y = 0$.
Q43 Sequences and Series Power Series Expansion and Radius of Convergence View
To each function $f \in E$, we associate the endomorphism $U$ of $E$. Let $\lambda \in \mathbb{R}^*$ be an eigenvalue of $U$ with eigenvector $f$, which is a solution of $(E_{1/\lambda})$. We assume that $f$ is developable as a power series on $\mathbb { R } _ { + } ^ { * }$, that is, there exists a power series $\sum _ { n \geqslant 0 } a _ { n } x ^ { n }$ of infinite radius of convergence such that $$\forall x \in \mathbb { R } _ { + } ^ { * } , \quad f ( x ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } x ^ { n } .$$ Using the results of Part IV, show that the only possible eigenvalues of $U$ are of the form $\lambda = 1 / p$ with $p \in \mathbb { N } ^ { * }$.
Q44 Differential equations Higher-Order and Special DEs (Proof/Theory) View
For $p \in \mathbb{N}^*$, let $P$ be a non-zero polynomial solution of $(E_p) : x(y'' - y') + py = 0$. It has been shown that $U(f)$ satisfies $y'' - y' = -f(x)/x$ and that $U$ is self-adjoint ($\langle f | U(g) \rangle = \langle U(f) | g \rangle$). Prove that the function $pU(P) - P$ satisfies on $\mathbb { R } _ { + } ^ { * }$ the differential equation $y ^ { \prime \prime } - y ^ { \prime } = 0$.
Q45 Differential equations Eigenvalue Problems and Operator-Based DEs View
For $p \in \mathbb{N}^*$, let $P$ be a non-zero polynomial solution of $(E_p) : x(y'' - y') + py = 0$. It has been shown that $pU(P) - P$ satisfies $y'' - y' = 0$ on $\mathbb{R}_+^*$. Show that $P$ is an eigenvector of $U$ for the eigenvalue $1/p$.
Q46 Continuous Probability Distributions and Random Variables Verification of Probability Measure or Inner Product Properties View
For all integer $p \in \mathbb { N } ^ { * }$ and all $x > 0$, we set $P _ { p } ( x ) = x \mathrm { e } ^ { x } g _ { p } ^ { ( p ) } ( x )$, where $g _ { p } ( x ) = x ^ { p - 1 } \mathrm { e } ^ { - x }$. We recall that $P _ { p }$ is a polynomial function of degree $p$, that $P _ { p } \in E$, and that $P_p$ is an eigenvector of $U$ for the eigenvalue $1/p$. The inner product on $E$ is $\langle f \mid g \rangle = \int _ { 0 } ^ { + \infty } f ( t ) g ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Show that the polynomials $P _ { p }$ are pairwise orthogonal in $E$.