Why Laplace transforms are so useful

Studying the forced harmonic oscillator by taking a Laplace transform and studying its poles.
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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

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https://3b1b.co/faq#manim

Music by Vincent Rubinetti.
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u

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