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Brilliant

Differential Equations I

via Brilliant

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Overview

Differential equations show up in just about every branch of science, including classical mechanics, electromagnetism, circuit design, chemistry, biology, economics, and medicine.

From analyzing the simple harmonic motion of a spring to looking at the population growth of a species, differential equations come in a rich variety of different flavors and complexities.
This course takes you on a grand tour of some of the most important differential equations of the natural sciences, giving you the hands-on experience needed to master the essentials.

Syllabus

  • Introduction: Discover why differential equations are so important!
    • Differential Equations in a Nutshell: What are differential equations and what are they used for?
    • Our First Equation: Use Newton's Law of Cooling to build and solve your first differential equation.
  • First-Order Separable Equations: Build and explore real-world models with first-order equations.
    • Modeling: The Drag Equation: Make a differential equation through experimentation.
    • Separate and Integrate: Conquer first-order separable equations with a single technique.
    • Application: Molecular Motor: Unravel secrets of the biological micro-world with first-order equations.
    • The Phase Portrait: Learn to sketch integral curves like a pro by solving this chemistry challenge.
    • Concavity and Partial Fractions: Improve your solution curves and add an important integration skill to your toolbox.
    • Application: In the Chem Lab: Practice your first-order skills while exploring this chemistry application.
    • Capstone: Vampires of Cancelvania: Use differential equations to save the countryside from a vampire plague!
  • Advanced First-Order Equations: Learn the power of combining geometry and differential equations
    • Direction Fields: Use a simple, geometric technique to draw integral curves on the phase plane.
    • Integrating Factor: Combine vectors and geometry into a technique for solving first-order problems.
    • Application: A Mixing Problem: Apply the integrating factor to a classic real-world problem.
    • The Potential: Explore the meaning of the integrating factor in the context of physics.
    • Application: Fluid Flow: Explore the potential for using first-order equations to make real-world models.
    • Capstone: The Great Escape: Drop in on a marooned astronaut who uses first-order equations to call for rescue.
  • Basics of Linear Systems: Matrices and vectors unlock the gateway to higher-order equations.
    • The Phase Plane: Learn to visualize the essential features of equation pairs.
    • The Matrix Exponential: Learn how to use vector geometry as a powerful tool for exploring linear systems.
    • Application: Underdamped Springs: Gain hands-on experience with the matrix exponential in a familiar setting.
    • Non-Diagonalizable Matrices: What do you do when you just don't have enough eigenvectors?
    • Review: Math of a Salesman: Practice your linear systems skills on a mind-altering robot sociology problem.
    • Nonhomogeneous Systems: Part I: Make an atom smasher by combining physics with systems of equations.
    • Nonhomogeneous Systems: Part II: Make a master formula for linear systems and complete your cyclotron.
    • Challenge: Floquet Theory: Strengthen your grasp on linear systems while learning how medical NMRI works.
  • Higher-Order Equations: The workhorse of physics, engineering, and applied mathematics.
    • Equations of Order Two: Uncover the deep connection between higher-order equations and systems.
    • Application: RLC Filter: Find out what second-order equations can do for electrical engineers.
    • Challenge: Higher-Order Equations: Reach beyond second-order and uncover a master formula with the Wronskian matrix.
    • Application: Hangin' Around: Solve a structural engineering problem with a third-order equation.
    • Application: Beam Me Up!: Can a building withstand an earthquake? Find out with fourth-order equations!
    • Application: Get Your Motor Runnin': Retrieve a treasure from the briny depths with a third-order equation.
    • Challenge: Why's The Sky Blue?: Answer a very common science question by applying the perturbation method.

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