Dive into a comprehensive introduction to Ordinary Differential Equations (ODEs) through this 4.5-hour video playlist. Explore key concepts including the definition of differential equations, separation of variables, Newton's Law of Cooling, and geometric interpretations. Learn about existence and uniqueness theorems, linear differential equations, and integrating factors. Discover techniques for solving various types of ODEs, including autonomous equations, Bernoulli equations, and constant coefficient homogeneous equations. Investigate mechanical vibrations, undetermined coefficients, and variation of parameters. Conclude with an introduction to solving ODEs using infinite series methods, covering ordinary and singular points, as well as specific examples like Airy's Equation.
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
