Introduction to Differential Equations

Created by: University of Waterloo, powered by Möbius

Description

The Introduction to Differential Equations Content Pack is a Möbius course developed by the University of Waterloo that you can use as a customizable starting point to a complete differential equations course in Möbius. This Content Pack covers standard approaches to solving first order ordinary differential equations along with associated qualitative analyses, second order linear differential equations, and systems of differential equations. This customizable resource contains 7 units of sectioned lessons and assignments enhanced with Möbius capabilities including in-lesson questions, interactive narratives, Math Apps, immediate feedback, and end-of-section assignments.

How does Möbius take the University of Waterloo’s content to the next level?

  • Lessons contain interactive elements like Interactive Narratives, HTML objects, Math Apps, or Geogebra applications to help solidify difficult STEM concepts.

  • Learn how Möbius’ unique STEM question types throughout this content provide the best STEM learning experience.

  • Work with over 50 configurable assessment properties when modifying existing or building your own assessments.

Course Structure

  • All content is organized into 7 units for easy navigation
  • Course materials provide a solid foundation for creating your course offering which include 50 lessons with illustrative visualizations, unlimited practice, Math Apps, interactive narratives, and immediate feedback to promote Active Learning
  • Different forms of assessment materials are distributed across 11 assignments to evaluate student comprehension through a variety of different question types
  • Pull from a vast selection of over 640 questions that you can use to create your own lessons and assignments or supplement existing ones
7 units
50 lessons
11 assignments
640+ questions

Unit 1: Introduction to Differential Equations

  • 1.1 What is a Mathematical Model?

  • 1.2 Models Galore with a Simple Linear ODE

  • 1.3 Classification of Differential Equations and Solutions

  • 1.4a Models for Motion With Gravity and Drag

  • 1.4b Motion of a Projectile

Unit 2: First order DEs

  • 2.1 Integrating Factors and Solution Structure for Linear First Order DEs

  • 2.2 Superposition of Multiple Inputs to Linear First Order DEs

  • 2.3 Undetermined Coefficients for Linear First Order ODEs

  • 2.4 Linear First Order Models: A Lethal Mix

  • 2.5 Linear First Order Models: Sticky Marbles

  • 2.6 Linear First Order Models: Rockets and Raindrops

  • 2.7 Solving Separable First Order DEs

  • 2.8 Separable First Order Models: The Skydive

  • 2.9 Separable First Order Models: Logistic Growth

  • 2.10 Solving Exact Differential Equations

  • 2.11 Solving First Order DEs by Transformations

  • 2.12 Solving First Order DEs by Reduction of Order: Chains and Cables

  • 2.13 Solving DEs by Reduction of Order: Escape Velocity

  • 2.14a Existence and Uniqueness for Linear First Order ODEs

  • 2.14b Existence and Uniqueness for First Order DEs in General

Unit 3: Qualitative Methods

  • 3.1 Direction Fields and Qualitative Graphs

  • 3.2 Qualitative Analysis and Stability of Equilibrium for Autonomous DEs

  • 3.3 Dimensional Analysis: Measurements and Dimensional Principles

  • 3.4 Dimensional Analysis: Dimensionless Variables

  • 3.5 Dimensional Analysis: Dimensional Matrices and the Buckingham Pi Theorem

  • 3.6 Dimensional Analysis: Using Dimensional Analysis to Explore the Unknown

Unit 4: Linear second order DEs

  • 4.1 Solutions of ay''(t) + by'(t) + cy(t) = 0

  • 4.2 Theory for the IVP y''(t) + p(t)y'(t) + q(t)y(t) = 0, y(t0) = y0, v(t0) = v0

  • 4.3 Applications of y′′(t) + p(t)y′(t) + q(t)y(t) = 0

  • 4.4 Finding Solutions for y′′(t) + p(t)y′(t) + q(t)y(t) = g(t)

  • 4.5 Applications of y′′(t) + p(t)y′(t) + q(t)y(t) = g(t) : Forced Oscillations

  • 4.6 Applications of y′′(t) + p(t)y′(t) + q(t)y(t) = g(t) : Amplitude Resonance

  • 4.7 Applications of y′′(t) + p(t)y′(t) + q(t)y(t) = g(t) : Beats

  • 4.8 Variation of Parameters and Transformations

Unit 5: Laplace Transforms

  • 5.1 Introduction to the Laplace Transforms

  • 5.2 Existence Conditions and Heaviside Functions

  • 5.3 New Transforms from Known Laplace Transforms

  • 5.4 Laplace Transforms of Derivatives

  • 5.5 Inverse Laplace Transforms

  • 5.6 Solving IVPs with Laplace Transforms

  • 5.7 Solving IVPs with Discontinuous Input

  • 5.8 Response to Impulsive Input

  • 5.9 Convolution of Functions

Unit 6: Linear Vector DEs (Systems of DEs)

  • 6.1 Brief Review of Matrices, Eigenvectors, and Solving Ax = b

  • 6.2 Introduction to Linear Systems of DEs

  • 6.3 Solving the Homogeneous System

  • 6.4 Fundamental Matrices and the Matrix Exponential

  • 6.5 Solving Systems by Laplace Transforms

  • 6.6 Phase Portraits of Vector DEs

  • 6.7 Solving Inhomogeneous Linear Systems

Quizzes

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