Laplace Transform Full Guide

Desmos graphs:

[ArduinoRR]

I remember this from college electronics. At first I was stupified. Then I realized the magic of transforming a problem that is difficult to solve into one that it easier to solve. Brilliant, thank you Laplace! And thanks for the fine video!

[jimcoughlin]

This was an excellent review for those of us who are years from our DiffEQ courses. I have always thought that all of higher mathematics consists of taking a problem in one space, transforming it to another, solving the problem in the new space, and then transforming it back. There are of course all sorts of complications with this and under what conditions the inverse transformation exists. But the Laplace transform, like the Fourier transform are great examples of this. Once you understand this, even using functors in Homological algebra is quite straightforward.

[digguscience]

Previously got this material when analyzing system stability, and transient effects. Laplace transform is a cool mathematical analysis.

[carultch]

The Laplace of the sine of t
Times that of t squared plus 3
Minus one over s
And invert the whole mess
Is t squared minus cosine, you see

[F.E.Terman]

Héél netjes uitgelegd! Gelukkig is in ons vak alles exponentieel (c.q. sinus) 🎉.

[radiac]

There does exist a definition of the inverse Laplace Transform as a definite integral. Unfortunately, it is a complex contour integral and treats frequency as a complex number (the real part is conventional frequency we all know, the imaginary part has no physical meaning, but is necessary to do the math). That said, electrical engineers always ignore this and just use tables to look up inverses, because they don't get paid to do integrals, they get paid to design circuits that work (/sarcasm). The function that zeroes the original function for negative values of time has a name, it's called the Heaviside Stepfunction, named after English telegraph engineer and self-taught math whiz Oliver Heaviside. It may seem a bit odd to name such a simple function after a person, but interestingly, the Heaviside Stepfunction is closely related to one of the most famous functions of all time, the Error Function.

[ardayazar]

Wish I saw this before my circuit finals 😂

[mohammadjadallah]

I beg anyone please tell me why at 4:15 the "solution" is the sum? I know if you have a quadratic formula
(x-1)(x-2) = 0, then x MUST be EITHER 1, OR 2. X=1, 2. However, here you don't explain why it's the SUM between the 2 different functions? Basically you're saying that x = 1+2, which for me makes no sense.

I am aware that the real factor is
(D-1)(D-2)v = 0 and the D's aren't really representing "x" and v is being multiplied at the outside but am soo eager to understand why it's the sum. I encountered this in a lecture long ago as well and never understood why. It was never explained to me.

If the explanation is too long, it's okay for anyone reading this not to explain it, but to provide me with where to find a proper explanation.

If the explanation is easy I beg anyone to explain it to me, thank you so much guys!!

[Omar_MTH]

I'm gonna pretend that I didn't think it is solvable by the quadratic equation solution

[stefanalecu]

It was a good video, but please improve your mic