Laplace transform is a linear operator

now we're going to prove one of the fundamental properties of the operator laas transform so we assume that we have two functions FS1 and FS2 and we assume that the laas transforms exists and this case we want to prove that the lapas transform of the combination is the combination of the the laas transform we will need this one when we have some equation differential equations like this y second + y Prime is equal to some sin t e to T okay and we want to solve this equation here we can take the laas transform of these two and we can take them after that one by one so we can take laplus of Y second and after to that laas of Y Prime and this one this is a product we're going to talk about that one later okay so this explains why we needed this one and to prove this result here we we assume that uh uh C lapas of fub1 is the integral between Z and Infinity of C1 e minus St fub1 T DT this one exists for some s bigger than S1 and C2 plus FS2 is the integral between Z and Infinity of C2 e to minus s F2 DT this one exists for some s bigger than S 2 and we see that uh if we add this two C1 L plus of fub1 + C2 L plus of FS2 we get this integral of e to minus St C1 F1 plus C2 F2 DT by taking s bigger than both S1 and S2 okay and this means that this one here is just the laas of C1 fub1 plus C2 F2 okay and this means that we can talk about linearity okay this proves that we when we dealing with differential equations like this as I said earlier uh y second or Y Prime + y equal to some e to T okay so we can take the lapas of Y Prime + y and laas of e we know this one and we can here linear we can take laas of Y prime plus laas of Y is equal to laas of e so here this is why we need to this result here now we're going to talk about this one here so we don't know the laas of a derivative of a function okay that's going to be the aim next and after that we're going to talk about La plus of two functions the product of two functions so we can get all the results that we need to be able to solve a differential equation so the aim of all these properties is to be able to solve a differential equation okay next we're going to prove results like this so we have laas of some derivative okay some function that's the the ant derivative of some function what's that in terms of the lapas of the other derivatives okay we will see that and we're going to use it in some equations like this thank you

Channel: Archimedes Notes

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Id: bn1PZwY0Tjo

Length: 4min 8sec (248 seconds)

Published: Fri Jul 12 2024