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Solve the initial value problem below using the method of Laplace transformsy" - 2y' 1Oy = 54 e 41, y(0) = 3,Y' (0) = 15Click here_to view the_ table...

Question

Solve the initial value problem below using the method of Laplace transformsy" - 2y' 1Oy = 54 e 41, y(0) = 3,Y' (0) = 15Click here_to view the_ table_of Laplace transforms Clckhere_to_view the_table_oLpreperties of Laplace_transtoms_y(t) = (Type an exact answer in terms Of € )

Solve the initial value problem below using the method of Laplace transforms y" - 2y' 1Oy = 54 e 41, y(0) = 3,Y' (0) = 15 Click here_to view the_ table_of Laplace transforms Clckhere_to_view the_table_oLpreperties of Laplace_transtoms_ y(t) = (Type an exact answer in terms Of € )



Answers

Use the Laplace transform to solve the given initial-value problem. $$y^{\prime \prime}-3 y^{\prime}-4 y=4 e^{-t}, \quad y(0)=1, \quad y^{\prime}(0)=1$$

Which himself? Differential equation. Health is differential equation. Why? Double prime? Why crying plus two y You're going four terms, Uh, three t And this is what? The initial condition? Where wise dio zero. Yeah. Why? Prime of zero. The repository farm. You should be able to get their self square times. Why? As mass to be tired s two's. Why has why four divided by as many as three. Why, uh, is Holt four fire I that as a means to James Mass three. We could be too. Remain Is four by minus two. It was Y minus one. You okay? Bye. Yes, three. Then how do a positive version your years ago I t just cool till no four e que e to e to be okay.

So for Why? Why you keep it, you five. And why weren't three year old hold Teoh One? So that'll be rearranged This plus tree Why, right? I notice one when you came Why? Who follows before? Why has equal to by s Square one Waas wanted to buy by this queer was four. Why I fine to t plus.

Trying to do the posturing form. Why Double prime? Why double place me prime pushed to He's going for war for a wise hero. Hong's Europe with I Prime born, not one. Why prime zero? He calls. It's a little beat doing posturing form square minutes three s Who's to why has multiple my likeness minus one. This one he pulls four divided by passed away Simplifying. Why is for I I That's raise one Delta pi pi as still and I get a parking back Fine divided by and it's one to us To this three. It s a little passing personally, whitey equal to me e e t plus.

All right. So this problem we're giving me of Y four prime minus four Y three prime plus six. Wide to prime minus four Y. Prime plus six plus Y equals zero. And we're giving that wires equals zero Y. F. Prime of zero equals one. Why double problems equals zero. And my show prime of zero equals one. I just want to get a little bit messy because of um Hi I have an order. This equation is but if we started here's what's gonna background why we're going to take the phosphorus from the of why for prime there and take four times and applause transform of for native four or three times off transform why three prime plus six times The Floss Transform of Y two Prime And the Native four times little applause transform. Uh let me rest this. Yeah, it's freezing Native forest on the pla transform of why prime plus the question why equals zero. All right, so this next equation is gonna get a little messy So I'm going to have to invent So the indented line is continued from this equation. So I have that s to the 4th round of applause transform of why my ass s cubed times Y zero clients essence square times Y. Prime is zero minus S times Y. Double prime of zero. My ass Why triple prime is your because that is the blast transform of why four prime? And I'll have um minus four let's put up so proud this for now we'll distribute in a second. Sorry let me area since I've got the s term Last four different cities and insider S. Cube a little fast transform of why minus S squared Y zero my ass S.Y. Prior zero. So right and so we'll continue into the second line now so I'm gonna get lost at the some down here now minus Why don't perform a zero. And the principal don't have plus six awesome. Keep that to the princess for now as well as supply things. S square the past transform of Y my S. zero Sir No no loss transfer Nice. S. Times quiet zero. My S. Y. Prime is zero. Then we're gonna have -4 times. S Fox transform of why I was going to factor this out right now so minus four. Uh we have a native four times uh a negative sometimes my so we're gonna anyway so doesn't attitude plus four Y zero and then we'll have plus plus transform of what equals there. Alright so now in order to you know make this simpler, maybe easy to read or not to cancel us in terms So in red or have these two terms. So we see that Y00 I'm going to go through and find every term that has a wide zero in it and we know this whole thing will cancel out. So why is there It's gonna go zero so all those terms cancel it to zero. Now we know the double Y double prime. Zero also equals 00. And search for these terms one here because zero, we have one here that goes there. Those are only two terms. Mhm. Now in green we have to uh we have water primary because there was one And why? Triple Prime is there was one. So we know that since those evil one, whatever they're multiplied to just equal that value. So this here is gonna be the one this year ago. Warm this to one. Um this will go to one and that's all our terms. Now we can rewrite it and be much simpler and easier to read. Just for me going to be like this As to the 4th little lost transformed over why? So our first term is cancelled out. It's gonna have minus S squared. Um I guess one -4. I'm gonna keep it outside the principal national distributing second sq plasma some of y minus S. Yeah Plus six kW in France as well. S where I was lost transforming why? Last one? My ass four S plus transforming Y plus applause transform of why equals zero. Alright, so now we're gonna now we've written this out just a little bit easier to read. We're gonna distribute these numbers we have outside of Princes. Make this equation again so that we can simplify As to the 4th loss transfer of Y. Was wanting mine s squared minus one -4. Sute. was the classrooms from over Y plus for S us six s. Fossil. Why last six miles for as classrooms from of y plus lots lots of what he pulls zero. Alright, so now we're going to bring all the terminals aren't attached to LA paz transform over to the right side like usual. And then we're gonna the fact that the boss transform the remaining terrorists. So that leaves us with applause transfer. Why Won't fly by us to the 4th -4 s huge plus six S squared My S four s plus one. All right, Harold. We're I think we're crowded at the end. It was my Ass four ass class one equal to all in terms of S squared plus one minus for us plus six, there's also going to be equal to S squared plus one -4. I'm Sorry. Mhm. I was writing the same thing and S squared minus for us plus seven and that is what we are what we have remaining. Um So school down more ranches were now trying to isolate little Bosman's himself. Applause of Y is equal to S squared minus four S plus seven all over As to the 4th last for as cute plus six S squared -4 s Plus one. Yeah. So now this obviously I'm thinking does not look like any of the local office transform that we're comfortable with our familiar with anything. So we're gonna try to do partial fractions with the partial fraction decomposition. So first we're gonna need a breakdown that denominator into some of that is easier to handle. Mhm. And I'm not going to do all these calculations are whatever on uh in this video but if you'd like to do it, this whole thing down here factors out to be um our time in our first two this week s squared last four S plus seven and are denominated down here is equal to s minus one to the fourth. Now we can do it that is a separate this out into our partial fractions. So we have four totals and four through A. Over. Yeah That's my last one plus B. Over S -1 Squared Plus C. Over S -1. Cute plus D. Over s. s. one so forth. So it's gonna come out he was a. times S -1. Cute plus B. Times S -1 Squared Plus C. I was S -1 plus d. So again I'm not gonna dry all these calculations with or whatever but this is going to come out to is going to be a as cute minus three A. S squared Plus three A. S minus A. I'm sorry then. Mhm minus. Okay now we're gonna have to be turned around plus Bs where minus two Bs plus B plus C. S minus C. Plus the I'm gonna say it's legal to our former mayor which s squared My last four ass plus seven. So now we'll separate this out into like terms. So for a s cube that is going to be equal to zero because we have no cubes in our other on the right hand side of this equation now have now we'll go for the s squares so yeah 93 A. S squared plus B. S squared. Now I'll be able to ask squared now for our justice solo ss of three A. S. Uh I'm sorry minus two Bs plus CS equals negative for S. And then our terms with no ss will be negative A. Plus the my N. C. Plus D. Equals seven. All right so now cancels out here A. Is equal to zero. So if this whole thing is zero divided by these And b. is equal to one. So every equation A. Is going to zero and B. You will go to one. This is now if you cancel this to file anything by s never have negative to be as wants to finish with this negative two plus C. People's negative for. And over here we're gonna have one minus C. Plus the eagles seven. So see here is going to be equal to negative four plus two which is just two. I'm sorry thanks to. All right and now we're going to bring that over here so I have one plus two plus The equal seven. So d. is going to be seven -3 D. is equal to four. Sweet. So now we have all our terms we can put them back into our partial fractions. Mhm. So I'm just gonna not right the aids because it's gonna be easier it's gonna be nothing um silent. Mhm. Quick stop war. So I have one over. Yes my as one square and let her beat her minus C. Turned to over S -1. Cute plus for over S -1 to the 4th Root. I have those terms we're gonna take a little applause transform so we're gonna have that oh uh y is equal to the inverse laplace transform of one over. Yes my as one squared uh minus. And do you uh inversion applause transform of um To over S -1? That's my as well in squared cubed. Actually I'm sorry Beach and then plus embarrassed applause transform of four over S minus one to the fourth. Give me some indulgent. My work before we continue this zoo is all right. Yeah this is looking good. All right now we recognize these uh applause transforms or we should anyway uh as being the yellow floss transforms for um uh T to the end Times The Exponential Function Times 8 to the T. Where these will be our this will be our mm uh This has to our exponential function and the term attached to our mm ah R. T. value is going to be our experience here -1 because we know that this this value or applause transform of T. to the n. e. to the 80. Alright that's easier to explain. is you go to uh huh. It. Uh and over uh sorry I'm blanking playing for words right now but uh S. Y. S. A. To the and plus one. Um I'm telling flight on that word mm And factorial. So it's in factorial over S. My S. A. To the end of one. So that leads us to see that this first one is going to be Y. Is equal to um 2 to the one E. To the won T. So I'm going to look at excellent is going to be mine T. Squared she. E. To the T. So the next one is a little bit trickier Because there are four up there were four other numerator And a four here. This was before here we know that this of top should be three factorial which is equal to six which means we have to take out a 4/6 Uses six. Still have Plus 4/6 um T. to the 3rd sometimes E. To the T. So it's a long and kind of messy um route but that is a solution that we end with


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