3.0 A better way to understand Differential Equations | Nonlinear Dynamics | Linearization

Part 1: https://youtu.be/ySfs8YVMY7Q Part 2: https://youtu.be/I9UEBRya4X0 Part 3: https://youtu.be/6mLCFyEv3Z0 These second-order nonlinear differential equations can be written in the form: dx/dt = f(x,y) dy/dt = g(x,y) Got a nonlinear differential equation? No problem, just linearize it! This method approximates the vector field as a linear equation at fixed points. This is very helpful for understanding stability and giving some qualitative ideas about the 'big picture' dynamics. There are draw-backs to linearization though. Firstly, it only illuminates the flow around the fixed points. Secondly, special types of fixed points that weren't covered in this series (centers, stars etc) cannot be reliably predicted using this method. Chapters: 0:00 Intro & Recap 0:31 Motivation 1:14 Damped Pendulum Equation of Motion 2:20 Damped Pendulum Phase Plane Generation 3:02 Damped Pendulum is Nonlinear 3:32 1D flow Linearization Motivation 3:57 2D flow Linearization Maths 4:29 Damped Pendulum Phase Plane Fixed Points Classification 5:44 Stable Spiral Initial Condition 5:55 Saddle Initial Condition 6:09 Van der Pol Phase Plane 6:38 Van der Pol Linearization 7:09 Rambling about Nonlinear Differential Equations 7:47 Outro Music: Music by Vincent Rubinetti Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u Other Music: Nuclear Lynx - Discovery -- Patreon: https://www.patreon.com/VirtuallyPassed Instagram: https://www.instagram.com/virtuallypassed/