Differential equations - Why Laplace transforms are so useful

Studying the forced harmonic oscillator by taking a Laplace transform and studying its poles. Instead of sponsored ad reads, these lessons are funded directly by viewers: https://3b1b.co/support An equally valuable form of support is to simply share the videos. Home page: https://www.3blue1brown.com Chapter on the Laplace Transform: https://youtu.be/j0wJBEZdwLs Chapter on the S-plane and Simple Harmonic Motion: https://youtu.be/-j8PzkZ70Lg Timestamps: 0:00 - Opening puzzle 1:06 - Key properties of a Laplace Transform 3:29 - Qualitative analysis with Laplace Transforms 4:29 - The Laplace Transforms of a Derivative 6:06 - The forced oscillator 11:59 - Intuition from the transformed solution 15:15 - Inverting to find a final answer 17:40 - Explaining the derivative property ------------------ These animations are largely made using a custom Python library, manim. See the FAQ comments here: https://3b1b.co/faq#manim Music by Vincent Rubinetti. https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. If you're reading the bottom of a video description, I'm guessing you're more interested than the average viewer in lessons here. It would mean a lot to me if you chose to stay up to date on new ones, either by subscribing here on YouTube or otherwise following on whichever platform below you check most regularly. Mailing list: https://3blue1brown.substack.com Twitter: https://twitter.com/3blue1brown Bluesky: https://bsky.app/profile/3blue1brown.com Instagram: https://www.instagram.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Facebook: https://www.facebook.com/3blue1brown Patreon: https://patreon.com/3blue1brown Website: https://www.3blue1brown.com