This course on Ordinary Differential Equations (ODEs) aims to introduce students to 1st and 2nd order differential equations. By the end of the course, learners will be able to understand key definitions of differential equations, solve linear and nonlinear ODEs using various methods such as separation of variables, integrating factors, and series solutions. The course covers topics like Newton's Law of Cooling, autonomous equations, undamped mechanical vibrations, and series solutions for ODEs. The intended audience for this course includes students and professionals interested in gaining a foundational understanding of ODEs and their applications.
Overview
Syllabus
What is a DIFFERENTIAL EQUATION?? **Intro to my full ODE course**.
The Key Definitions of Differential Equations: ODE, order, solution, initial condition, IVP.
Separation of Variables // Differential Equations.
Newton's Law of Cooling // Separable ODE Example.
The Geometric Meaning of Differential Equations // Slope Fields, Integral Curves & Isoclines.
The Big Theorem of Differential Equations: Existence & Uniqueness.
Linear Differential Equations & the Method of Integrating Factors.
The Method of Integrating Factors for Linear 1st Order ODEs **full example**.
The Bernoulli Equation // Substitutions in Differential Equations.
Autonomous Equations, Equilibrium Solutions, and Stability.
The Logistic Growth Differential Equation.
The Theory of 2nd Order ODEs // Existence & Uniqueness, Superposition, & Linear Independence.
How to Solve Constant Coefficient Homogeneous Differential Equations.
Constant Coefficient ODEs: Real & Distinct vs Real & Repeated vs Complex Pair.
Higher Order Constant Coefficient Differential Equations: y'''+y'=0 and y''''-3y'''+3y''-y'=0.
Linear Independence of Functions & The Wronskian.
The Theory of Higher Order Differential Equations.
Undamped Mechanical Vibrations & Hooke's Law // Simple Harmonic Motion.
Mechanical Vibrations: Underdamped vs Overdamped vs Critically Damped.
Undetermined Coefficients: Solving non-homogeneous ODEs.
Variation of Parameters || How to solve non-homogeneous ODEs.
How to solve ODEs with infinite series | Intro & Easiest Example: y'=y.
When can you use Series to solve ODEs? Ordinary vs Singular Points.
How to use SERIES to solve DIFFERENTIAL EQUATIONS example: Airy's Equation y''-xy=0.
Taught by
DrTreforBazett
