If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content
Current time:0:00Total duration:11:36

Laplace as linear operator and Laplace of derivatives

Video transcript

well now's as good a time as any to go over some interesting and very useful properties of the Laplace transform and the first is to show that it is a linear operator what does that mean well let's say I wanted to take the Laplace transform the Laplace transform of the sum of that we call it the weighted sum of two functions so say some constant c1 times my first function f of t plus some constant c2 times my second function G of T well by the definition of the Laplace transform this would be equal to the improper integral from 0 to infinity of e to the minus s T times whatever our our function that we're taking the Laplace transform of total times c1 F of T plus c2 G of T I think you know where this is going all of that DT and then that is equal to integral from 0 to infinity let's just distribute the e to the minus s T that is equal to what that is equal to c1 c1 e to the minus st f of t plus c2 e to the minus st g of t and all of that times DT and just by the definition of how or the properties of integrals work we know that we can split this up into two integrals right if the the integral of the sum of two functions is equal to the sum of their integrals and these are just constants so this is going to be equal to c1 times the integral from zero to infinity of e to the minus st times f of t DF t plus c2 times integral from zero to infinity of e to the minus s T G of T DT and this was just a very long-winded way of saying well you know what is this this is the Laplace transform of F of T this is the Laplace transform of G of T so this is equal to c1 times the Laplace transform of F of T plus c2 times this is the Laplace transform the Laplace transform of G of T and so we have just shown that the Laplace transform is a linear operator right the Laplace transform of this is equal to this so essentially you can kind of break up the sum and go and you know take out the constants and just take the Laplace transform that's something useful to know and you might have guessed that that was the case anyway but now you know for sure now we'll do something which I consider even more interesting and and this is actually going to be a big clue as to why the plus transforms are extremely are extremely useful for solving differential equations so let's say I want to find the Laplace transform the Laplace transform of F prime of T so I have some F of T I take its derivative and then I want the Laplace transform of that let's see if we can find a relationship between the Laplace transform of the derivative of a function and the Laplace transform of the function so we're going to use some integration by parts here so let's let's just put the integration well actually let me just say what this is first of all this is equal to the integral from 0 to infinity of e to the minus st times F prime of T F prime of T DT and to solve this we're going to use integration by parts I let me write it in the corner just we remember what it is so I think I memorize it because I recorded that last video not too long ago that the integrals can write the shorthand the integral of u of you what's a UV prime because that'll match what we have up here better is equal to the both functions without the derivatives you V minus the integral of the opposite so the opposite is U prime V U prime V so here the substitution is pretty clear right because we don't want it we want to end up with we want to end up with a with f of X right so let's make V prime is f Prime and let's make you eat ooh the minus SD so let's do that U is going to be e to the minus st and V is going to equal what V is you going to equal F prime of T and then u prime u prime would be minus s e to the minus st and then and then V prime oh sorry this is V Prime right right V prime is f prime of T V prime is is F prime of T so V is just going to be equal to F of T hope I didn't say that wrong the first time but you see what I'm saying this is you that's U and this is V Prime and if this is V prime then if you take the antiderivative both sides and V is equal to f of T so let's apply integration by parts so this Laplace transform which is this is equal to UV which is equal to e to the minus s T this is equal e to the minus s T times V F of T F of t minus minus the integral minus the integral and of course we're going to have to evaluate this from 0 to infinity I'll keep the I'll keep the the improper integral with us the whole time I won't switch back and forth between the the definite and indefinite integral so - this part so the integral and from 0 to infinity of u prime u prime is minus s e to the minus s T times V V is f of T f of T DT now let's see we have a - a - let's make both of these pluses this s is just a constant so we can bring it out so that is equal to that is equal to e to the minus s T F of T evaluated from 0 to infinity or as we approach infinity plus s times the integral from 0 to infinity of e to the minus s T F of T DT and here we see what is this this is the Laplace transform of F of T right so this is equal to let's evaluate this part so when we evaluated infinity as we approach infinity e to the minus infinity is approach 0 F of infinity now this is an interesting question this is an interesting question F of infinity I don't know that could be large that could be small that could approach some that approaches some value right this approaches 0 so we're not sure if this if this increases faster than this approaches 0 then this will diverge I won't go into into the mathematics of whether this converges or diverges but let's just say in very rough terms that this will converge to 0 if F of T grows slower then e to the minus st shrinks and I'll maybe later on we'll do some more rigorous definitions of under what conditions will F of T actually will this expression actually converge but let's assume that F of T grows slower than e to the S T or it converges it would grow slower than this then this then this or diverges slower than this converge is another way to view it or this grow slower than this shrinks so if this grow slower than this shrinks then this whole expression will approach zero and then you want to subtract this whole expression evaluated at 0 so e to the 0 is 1 my times F of 0 so that's just F of 0 F of 0 plus s times we said this is the Laplace transform of F of T that's our definition so the Laplace transform of F of T and now we have an interesting property what was the left-hand side of everything we were doing the Laplace transform of F prime of T so let me just write it all over again so we now know that and I'll switch colors the Laplace transform of F prime of T is equal to s times the Laplace transform of F of t minus F of 0 and now let's just let's just extend this further what is the Laplace transform and this is a really useful thing to know what is the Laplace transform of F prime prime of T well we can do a little pattern matching here right that's going to be s times the Laplace transform of its antiderivative times the Laplace transform of F prime of T right this goes to this that's an antiderivative this goes to this that's one antiderivative minus F prime of 0 right but then what's the Laplace transform of this this is going to be equal to s times the Laplace transform of F prime of T but what's that that's this right that's s times the Laplace transform of F of T - f of zero right I just substituted this with this - f prime of zero and we get the Laplace transform of the second derivative is equal to s squared times the Laplace transform of our function f of t minus s times f of zero minus F prime of zero and I think you're starting to see a pattern here this is a little a strand form of F prime prime of T and I think you're trying to see why the Laplace transform is useful it turns derivatives into multiplication in two multiplications by s and actually as you'll see later it turns in terms integration - divisions by s and you can take you can take our [ __ ] derivatives and just keep multiplying by s and you see this pattern and and I'm running out of time but I'll leave it up to you to figure out what the Laplace transform of the third derivative of F is see you in the next video