Laplace transforms -- differential equations 18

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[MrFtriana]

If you are interested, you can find the inverse Laplace transform using complex variables methods, specifically, the Bromwich integral.

[s]

18:23 you saw it here first! (s+2)(s-1) = s^2-s+2 🤣 [If you expand it correctly, (s+2)(s-1) = s^2+s-2]
19:29 And he strikes again! 1/(s-1) goes to exp(+t) instead of exp(-t)

[jamesfortune]

Thinking of the operational aspect of the Laplace Transform transforming a set of differential equations into a set of algebraic equations, where's the insight of how the convolution operations relate to derivative relationships? That would be awesome.

[jaxoncr]

at 33:35 shouldn’t it be e^(-t) since we had s+1 or am i not following

[schweinmachtbree]

To talk rigorously about the inverse Laplace transform you should have mentioned that the Laplace transform operator is injective - in fact this isn't actually true if you consider the Laplace transform operating on all functions, because two integrable functions which are equal except on a set of lebesgue measure zero will have the same integral. a class of functions for which the Laplace transform is injective is the bounded continuous functions, for example

[joshuanugentfitnessjourney]

Ty so much for the best math channel on YouTube, I am a mathematics bachelor, but couldn't find a job that uses math, Do you know of any apps or webpages to help me stay sharp with math? Daily calculus, DIffeqs, proofs, statistical problems, ect?

[Jkauppa]

laplace transform of derivative gets you the integrand and the F(0) term is the +C term

[lexinwonderland]

Great lecture as always, but the one comment I've had on this series is that it only admits real numbers. A lot of the proofs you've given would be much more succinct and less piecemeal using complex numbers, for example Laplace transform of trig functions. It's been years since I took diff eq and I'm mostly watching because I love your delivery, but I remember as a student complex numbers making ALL of this make way more sense, and now years later as an electrical engineer it seems absurd NOT to use complex numbers when the Fourier transform (which is much simpler to visualize for most, I'd imagine) as the imaginary-only case of the Laplace transform. Again, ymmv, and obviously you're doing a great job both on the individual lectures and on the channel's goal, but I feel like it's worth mentioning at least SOME other students may understand better if things are also framed with complex numbers.

[iabervon]

It's a bit surprising to see that you can work with Laplace transforms with s being implicitly a real number. It's "supposed" to be a complex number, but I guess we could even treat it as a formal variable for the purposes of this course if we're going to handle our irreducible polynomials with the sin and cos formulas instead of getting complex roots.

It seems like Laplace transforms (like generating functions) use some mathematical machinery that isn't introduced in the prereqs for these courses, where you make an expression with a new variable, manipulate the expression, and recognize the result as the result of introducing the variable in a combination of other expressions. It feels like the curriculum is missing a section somewhere on formal variables and what is safe to do with them.

[holyshit]

18:43 in the denominator is +2 but should be -2
otherwise complete square is what we can do in real numbers
Convolution and Borel's theorem is missing
Laplace transform of derivative is simply integration by parts