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Biologie – Écologie (1)

Ecosystems

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The evolution of species

Biologie – Génétique (3)

From DNA to protein
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The immune system

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The cell
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Photosynthesis

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Hormonal regulation and homeostasis
The neuron and the nerve message

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L'auto-assemblage supramoléculaire : cages et bols métallo-organiques

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The mole and the amount of substance
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Cryptographie – Applications (4)

Strong authentication: TOTP and FIDO2
The blockchain: hashing, Merkle trees and proof of work
Steganography: hiding the very existence of the message
Passwords: salting, hashing, and key derivation

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Side-channel attacks
Attacks on RSA

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Homomorphic encryption

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The Diffie-Hellman key exchange
Public-key cryptography and RSA
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Randomness in cryptography
The one-time pad and perfect secrecy
Kerckhoffs's principle and secure design
Secret codes: encryption and decryption
One-way functions: the foundations
Vigenère and the early days of cryptanalysis

Cryptographie – Intégrité (3)

Hash functions and digital signatures
Hash functions: construction and attacks
MAC and HMAC: authenticating messages

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Modular arithmetic, the foundation of cryptography
Post-quantum cryptography

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End-to-end encryption
Signal: forward secrecy and the double ratchet

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Schnorr identification and the Fiat-Shamir transform
Secure multiparty computation
Shamir's Secret Sharing
Zero-Knowledge Proofs

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Quantum key distribution (BB84)

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Certificates and the chain of trust
TLS and HTTPS: web security

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AES, the Encryption Standard
Block ciphers and their modes
Modern symmetric encryption: AES and operating modes

Informatique – Algorithmique (6)

Introduction to dynamic programming
Algorithmic Complexity
Binary search
Recursion
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Circuits logiques et architecture des ordinateurs

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Les codes correcteurs d'erreurs

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Théorie de l'information et compression

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Functions
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How the Internet works

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Binary Trees
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Finite automata

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Discovering Algebra
The basics of the calculation

Mathématiques – Algèbre (13)

Expand an expression
First-degree equations
Quadratic equations
Factoring an expression
First-degree inequalities
Literal calculation
The second degree: canonical form
Notable figures
Matrices
Formulating problems as equations
Polynomials and factorisation
Sign of a trinomial
Systems of two equations

Mathématiques – Algèbre linéaire (6)

Linear maps
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Vector spaces
Linear systems and Gaussian elimination
Invertible matrices and solving systems
Eigenvalues and diagonalisation

Mathématiques – Analyse (31)

Analyse de Fourier et traitement du signal
Applications of differentiation
Bolzano-Weierstrass par dichotomie
Continuity and the Intermediate Value Theorem
A comparison of growth rates
Derivatives and variations
Limited developments
Calculus assignment: the function ln(1 + x)/x and its associated sequence
Differential equations
Integration by parts
L'aire entre deux courbes
L'intégration par changement de variable
The square function
The derivative
The exponential function
The inverse function
The natural logarithm function
The concept of a function
La valeur moyenne d'une fonction
Integral calculus
The derivative
Proof by induction
Le théorème de Heine
Le théorème de Rolle et les accroissements finis
Le théorème du point fixe de Banach
Affine functions
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Function limits
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Primitives of a function
Direction of variation of a function

Mathématiques – Analyse numérique (1)

L'intégration numérique

Mathématiques – Arithmétique (14)

Bézout, Gauss and Diophantine equations
Divisibility, multiples and divisors
Assignment: the Josephus problem
L'irrationalité de e
Fractions: meaning and representation
Decimal numbers
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Mathématiques – Géométrie (12)

Areas and perimeters
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Volumes of common solids

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L'argument diagonal de Cantor
Logique et méthodes de démonstration

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Le théorème de d'Alembert-Gauss
Complex numbers: algebraic form
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Counting and combinatorics
The binomial distribution
The normal distribution
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Descriptive statistics
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Direction of variation of a sequence
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The Fibonacci sequence

Mathématiques – Trigonométrie (5)

Cosine and sine of a real number
Trigonometric equations
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The unit circle
Trigonometric formulas

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Applications of the scalar product
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Physique (2)

Quantities, units and measurement
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Physique – Électricité (14)

Charging and discharging a capacitor (RC)
The current intensity
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Ohm's Law
Electrical voltage
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Alternating current
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Oscillations in an RLC circuit
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Electromagnetic induction
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Maxwell’s equations
Electrostatic potential and energy

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Coulomb's law

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Fluid flow
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Physique – Mécanique (22)

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Physique – Mesure et méthode (6)

Significant figures and uncertainties
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The experimental approach
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Scientific notation and orders of magnitude

Physique – Ondes (2)

L'effet Doppler
Les ondes mécaniques et l'acoustique

Physique – Optique (9)

Dispersion and spectra
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The construction of images
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The propagation of light
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Refraction and Snell–Descartes’ laws
Thin lenses

Physique – Physique nucléaire (1)

La radioactivité et la désintégration nucléaire

Physique – Physique quantique (10)

L'effet photoélectrique
Quantum tunnelling
Wave-Particle Duality
Energy Quantization and Spectra
The black body and the birth of the quanta
The laser: stimulated emission and population inversion
Heisenberg's uncertainty principle
Spin and the Pauli Principle
Atomic orbitals and quantum numbers
Superposition, entanglement and qubits

Physique – Relativité (1)

La relativité restreinte

Physique – Thermodynamique (3)

The ideal gas law
La thermodynamique et les machines thermiques
Les transferts thermiques

Physique – Thermodynamique quantique (1)

L'autonomisation des systèmes quantiques et l'opérateur travail

272 / 272 courses

Library Differential equations

Nature of the roots

MCQ

For the equation y'' - 6y' + 9y = 0, what is the form of the general solution?

Possible answers

y = C_1 e^{3x} + C_2 e^{-3x}

y = \left(C_1 + C_2 x\right)e^{3x}

y = C_1 \cos(3x) + C_2 \sin(3x)

y = C_1 e^{3x} + C_2 x e^{-3x}

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