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I'll now introduce you to the concept of the Laplace transform and this is truly one of the most useful concepts that you'll learn not just in differential equations but really in mathematics and especially if you're going to go into engineer and you'll find that the Laplace transform besides helping you solve differential equations also helps you transform functions or or waveforms from the time domain to the frequency domain and and study and understand a whole set of phenomena but I won't get into all of that yet now I'll just teach you what it is Laplace transform I'll teach you what it is make you comfortable with the mathematics of it and then in a couple of videos from now well I'll actually show you how it is useful to to use it to solve differential equations will actually solve some of the differential equations we used we did before using the previous methods but we'll keep doing it and we'll solve more and more difficult problems so what is the Laplace transform well plus transform in the the d the notation is the L like Luverne from Laverne and Shirley it might be before many of your times but I grew up on that so actually I think it was even reruns when I was a kid maybe well anyway so Laplace transform of some function and here the convention instead of saying f of X people say f of T and the reason is is because in a lot of the differential equations or a lot of engineering you actually are converting from a function of time to a function of frequency and don't worry about that right now if it if it if it confuses you but Laplace transform of a function of T it transforms that function into some other function of s and how does it do that well actually let me just let me just do some mathematical notation that probably won't mean much to you so what is a transform well the way I think of it it's kind of a function of functions right a function will take what will take you from one set of well then what we've been dealing with one set of numbers to another set of numbers a transform will take you from one set of functions to another set of function so let me to find this the Laplace transform for our purposes is defined as the improper integral and I know I haven't actually done improper integrals just yet but I'll explain them in a few seconds the improper integral from zero to infinity of e to the minus st times f of t so whatever is between the Laplace transform brackets d t now that might seem very daunting to you and very confusing but i'll now do a couple of examples so what is the Laplace transform well let's say that f of T is equal to 1 so what is the Laplace transform of 1 the Laplace transform of 1 so if f of T is equal to 1 it's just a constant function of time well that will be equal to and let me rewrite well actually let me just substitute it exactly the way I wrote it here so that's the improper integral from 0 to infinity of e to the minus st e to the minus st times one here well I don't have to rewrite it here but there's a times 1 DT and I know that that's probably that infinity is probably bugging you right now but but we'll deal with that shortly actually let's deal with that right now this is the same thing this is the same thing as the limit and let's say let's say it's a approaches infinity it's a approaches infinity of the integral from 0 to a e to the minus s T DT so this is just so you feel a little bit more comfort with it you might have guessed that this is the same thing because obviously you can't evaluate infinity but you can take the limit as something approaches infinity so anyway let's take the antiderivative and evaluate this improper definite integral or this improper integral so what's the antiderivative of e to the minus s T with respect to DT well it's equal to what minus 1 over s e to the minus s T right if you don't believe me take the derivative of this you take minus s times that that would all cancel out and you just be left with e to the minus s T fair enough and we're going to take actually let me delete let me delete this here this is equal sign cuz I could actually use some of that real estate we are going to take the limit as a approaches infinity you don't always have to do this but this is the first time we're dealing with improper integral so I figured might as well remind you that we're taking a limit and we're going to evaluate we took the antiderivative now we have to evaluate at a minus the antiderivative evaluated 0 and then take the limit of whatever that ends up being as a approaches infinity so this is equal to the limit as a approaches infinity ok if we substitute a in here first we get we get minus 1 over s remember we're dealing with t the the the we took the integral with respect to t e to the minus s a right that's what happens when I put a in here - now what happens if when I put T equals zero in here so when T equals zero - comes e to minus s times zero this whole thing becomes 1 and I'm just left with minus 1 over s so I - minus 1 over s fair enough and then let me scroll down a little bit I wrote a little bit bigger than I wanted to but that's ok so this is going to be the limit as a approaches infinity of minus 1 over s e to the minus s a - minus 1 over s so plus 1 over s so what's the limit as a approaches infinity well what's this term going to do as e approach as a approaches infinity the exponent is going to get if we assume that s is great than zero and we'll make that assumption for now let's listen let me write that down explicitly let's assume that s is greater than zero so if we assume that s is greater than zero then as a approaches infinity what's going to happen well this term is going to go to zero right e to the minus a googol is a very very small number enough e to the minus Googleplex is a very even smaller number so then this e to the minus infinity is approaches 0 so this term approaches zero this term isn't affected because it has no a in it so we're just left with 1 over s so there you go this is a significant moment in your life you have just been exposed to your first Laplace transform eventually and I'll show you in a few videos there are whole tables of Laplace transforms and eventually we'll prove all of them but for now we'll we'll just work through some of the more basic ones but this can be our first entry in our Laplace transform table that the Laplace transform the Laplace transform of F of T is equal to 1 is equal to 1 over s notice we went from a function of T although obviously this one wasn't really dependent on T to a function of s I have about three minutes left but I don't think that's enough time to do another Laplace transform so I will save that for the next video see you soon