6: Laplace Transforms - Dissecting Differential Equations

Explanation of the Laplace transform method for solving differential equations. In this video, we go through a complete derivation of why every part of the Laplace transform method works the way it does!

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[blackpenredpen]

MU PRIME!!!! You just answered the question that I had when I first took diff eq over a decade Wow!!! This is an amazing and exciting video. I enjoyed every second of it!!

[omniyambot]

I'm here because you were mentioned by bprp. Man, it was so worth it.

[UnrealNine]

I have finally come across this video, it's been several years trying to answer what seemed the million dollar question and you answered it!

Thanks for showing this information, i wonder why this isn't more extended

[VibingMath]

O man! I feel like I re-learn Laplace transform today! Thanks for explaining step by step the purpose behind! Keep going in this direction!

[premdeepkhatri]

Very Good Explanation of Laplace Transform to Visualize.

[SaidVSMath]

Applause, applause, applause. Amazing. I never knew why in gods name we cared about this thing and why it worked, but now I get it. I GET IT. OH MY GOD. THANK YOU.

[PaperQuestion]

This was a very interesting intuition for why we use Laplace Transforms, since I've no idea how it worked beyond writing it off as "sorcery" when I was only taught the table of Transforms and Inverses.

Seeing the step at 8:00 has me curious now.
Most elementary functions we know of such as the trigonometric or polynomial functions are dwarfed by the exponential function's growth rate, which allows for the upper limit to go to infinity and result in a value of 0.
What if the function y(t) grows faster than e^(st)?
Does this mean there are some functions that can't be solved using Laplace transforms?

[efbya]

I wish I had this information while I was doing my engineering. I asked many math professors what’s going on. They could not explain anything beyond “it’s defined that way”. Great video. Videos like this will inspire younger generation to learn mathematics!

[TekCroach]

Very beautifully explained. Actually I do love to see the young teaching. Thanks! :)

[bartomiejpotaman]

Maaan, can’t thank you enough. It’s not often seen that the concept is actually explained in a way that you clearly see what was the reasoning behind it and not some stroke of some genius. That’s what’s maths all about. Once again thanks - you should be proud of this vid.

[thilaknakumaratunga]

I am so grateful for your guidance. I never understood this concept before, but now it all makes sense. Thank you for being an amazing teacher 😍

[Vandarte_translator]

Thanks! This is very useful for my differential equations course. By the way, now I know how to solve homogeneous and non homogeneous differential linear equations thanks to you and my teachers. 👍

[skerJG]

I love teaching math by walking through a sort of discovery/"play-around" process. This is the first I've seen for the Laplace transform - absolutely wonderful!

[sander_bouwhuis]

At 11:55 you gloss over the fact that you rewrite the function into a very useful form. That requires insight because in many of its other forms it is very hard to get back to what needed to be integrated to get you there.
At 15:56 = mind blown!
It is amazing what happens when someone actually explains WHY things work the way they work.
I really had an AHA! moment during the first half of the video! Thank you, thank you, thank you!

[PranavViswanathan]

this is one of the most intuitive explanations of the laplace transform. the fact that you are the same age as me and explaining this better than my prof is super humbling but this series is the only thing getting me through my winter diffEq course.

[refathbari]

YOU ARE I NEVER THOUGHT THERE WOULD BE SUCH A BEAUTIFUL, SIMPLE MOTIVATION FOR LAPLACE TRANSFORMS, BUT HERE YOU EXPLAINED IT SO

[erikdurfey]

Excellent presentation of the Laplace transform! Many thanks 👍

[d.jensen]

Thanks for explaining this in a way that means something! Numerical analysis was the salvation of my engineering career. Using numerical methods I found I could solve anything! I was using it as a crutch, but desperate times called for desperate measures. Your video is beginning to make an honest engineer out of me. ;)

[lossen]

Thank you so much for this! This is so clearly explained that even someone, who has just begun really learning calculus can follow through and understand the concepts presented.
Well done!

[chrisgreen_]

That's a really well explained video! I've used the Laplace transform method, but I don't think I've ever really thought about how I'd derive it if I hadn't seen it before.