Differential Equations I

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.

• 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.