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# Laplace transform to solve an equation

## Video transcript

I've been doing a ton of videos on the mechanics of taking the Laplace transform but you've been sitting through them always wondering what am i learning this for and now I'll show you at least in the context of differential equations and I've gotten actually a bunch of letters on on the Laplace transform what does it really mean and and all of that and those are excellent questions and you should strive for that it's hard to really have an intuition of the Laplace transform in the differential equations context other than it being a very useful tool that that that converts differential or integral problems into algebra problems but I'll give you a hint and if you want a path to learn it in you should pursue learn about Fourier series and Fourier transforms which are very similar to Laplace transforms and that'll actually build up the the intuition on what the what the frequency domain is all about anyway let's actually use a Laplace transform to solve a differential equation and this is one we've seen before so let me see let me here there it is okay so let's say the differential equation is y prime prime plus five times the first derivative plus 6y is equal to zero and you know how to solve this one but I just want to show you with a fairly straightforward differential equation that you could solve it with the Laplace transform and actually it you end up kind of having a characteristic equation and the initial conditions are y of 0 is equal to 2 and Y prime of 0 is equal to 3 now to you to use the Laplace transform here we essentially just take the Laplace transform of both sides of this equation so we get let me use a more vibrant color so we get the Laplace transform of Y the second derivative plus well we can say the plus transform of 5 times y prime but that's the same thing as 5 times on applause transform of Y prime Y prime plus six times the Laplace transform of Y and let me ask you a question what's the Laplace transform of zero let me do that in a so the Laplace transform of zero to the integral from 0 to infinity of 0 times e to the minus stdt so this is a zero in here so this is equal to zero so the Laplace transform of zero is zero and that's good because I didn't have space to do it at all another another curly L so what are the Laplace transforms of these things well this is where we break out one of the useful properties that we learned and that useful property let me write it over here because I think that's going to be some I'll need as much real estate as possible let me erase this so what is so we learned that the Laplace transform I'll do it here actually I'll do it down here the Laplace transform of F prime or we could even say Y prime is equal to s times the Laplace transform of Y minus y of 0 right we we prove that to you and this is extremely important to know so let's see if we can apply that so the Laplace transform of Y prime prime if we apply that that's equal to s s times the Laplace transform of well go from Y prime to Y you're just taking the antiderivative so putting the antiderivative Y of the second derivative we just end up with the first derivative minus the first derivative at 0 notice we're already using our initial conditions I won't substitute it just yet and then we end up with plus 5 times I'll write it every time just so that we did this just a weekend so plus 5 times the Laplace transform of Y prime plus 6 time's the Laplace transform of Y all of that is equal to zero so just to be clear all I did is I expanded this into this using this so how can we rewrite the Laplace transform of Y Prime well we could use this once again so let's do that so this over here I'll do it in magenta this is equal to s times what s times the Laplace transform of Y prime well that's s times the Laplace transform of Y minus y of 0 right I took this part and replace it with what I have in parentheses so minus y prime of 0 and now I'll switch colors plus 5 times once again the Laplace transform of Y prime we could use this again so 5 times s times the Laplace transform of Y minus y of 0 plus 6 times the Laplace transform I know space 6 times a little plus I'll do it another line plus 6 times the Laplace transform of Y all of that is equal to 0 I know this looks really confusing but we'll simplify right now we could get rid of this right here because we've used it as much as we need to so now we just simplify I know it's using the Laplace transform we didn't have to guess at a general solution or anything like that even when we did characteristic equation we guessed what the original general solution was now we're just taking applause transforms and let's see where this gets us so simplifying actually just want to make clear because I know it's very confusing so this so I rewrote this part as this and I rewrote this thing as this and everything else is the same but now let's simplify the math so s tight so we get s squared s squared times the Laplace transform of Y I'm going to write smaller I've learned my lessson minus s times y of 0 s times y of 0 let's substitute Y of 0 here y of 0 is 2 so s times y of 0 is 2 times s so 2's so I distribute that s minus y prime of 0 Y prime of 0 is 3 so minus 3 plus so we have 5 times s times the Laplace transform of Y so plus 5 s times the Laplace transform of Y minus 5 times y is 0 Y of 0 is 2 so minus 10 minus 10 right 5 times this is 2 right here so 5 times 2 plus 6 times the Laplace transform of Y all of that is equal to 0 now let's group our Laplace transform of Y terms and our constant terms and we should be hopefully getting someplace so let's see my laplace transform of Y terms I have this one I have this one and I have that one so what am I left with so let me factor out the Laplace transform of Y part so I get the Laplace transform of Y that's good because it's a pain to keep writing it over and over times s squared plus 5 s plus 5 s plus 6 plus 6 so those are all my Laplace transform terms and then I have my constant term so if see I have 1 St so minus 2 s minus 2 s minus 3 minus 10 is equal to 0 and what can we do here well let's well this is interesting first of all notice that the the coefficients on the Laplace transform of Y terms that those are that characteristic equation that we dealt with so much and that is hopefully to some degree second nature to you so that's a little bit of a clue and if just if you want some some very tenuous connections well that makes a lot of sense because the characteristic equation to get that we substituted e to the RT and the Laplace transform involves a very similar function but anyway let's go back to the problem so how do we solve this so we're and actually let me just give you the big picture here because it's a good point what I'm going to do is I'm going to solve this I'm going to say the Laplace transform of Y is equal to something and then I'm going to say boy what functions Laplace transform is at something and then I'll have the solution if that confuses you just wait and hopefully it'll make some sense from here until that point is just some fairly hairy algebra so let's let's scroll down a little bit just so we have some breathing room and so I get the Laplace transform of Y times s squared plus 5 s plus 6 is equal to let's add these terms to both sides of this equation is equal to 2 s plus 3 plus 10 oh that's silly plus 13 this is minus 13 here a phone call who's calling I think it's some kind of marketing phone call anyway 2's plus 13 and now what can I do well let's divide both sides by this s squared plus 5 s plus 6 so I get the Laplace transform the Laplace transform of Y it equal to 2's plus 13 over s squared plus 5 s plus 6 now we're almost done everything here is just a little bit of algebra so now we're almost done we we haven't solved for y yet but we know that the Laplace transform of Y is equal to this now if we just add this in our table of our Laplace transforms we would immediately know what Y was but I don't see something or I don't remember anything we did in our table that looks like this like this expression of s I'm essentially out of time so in the next video we're going to figure out what what functions Laplace transform is this and it actually turns out it's a sum of some of things we already know and we just have to manipulate this a little bit algebraically see in the next video