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Q1. Solve using Laplace Theorem the following system of ODEs. 2 Vz +6. 100 100100100 Vz-The inilial conditions are V1(0) = 0,Vz(0) = 150....

Question

Q1. Solve using Laplace Theorem the following system of ODEs. 2 Vz +6. 100 100100100 Vz-The inilial conditions are V1(0) = 0,Vz(0) = 150.

Q1. Solve using Laplace Theorem the following system of ODEs. 2 Vz +6. 100 100 100 100 Vz- The inilial conditions are V1(0) = 0,Vz(0) = 150.



Answers

Use Laplace transforms to solve each of the initial-value problems in Exercises $1-18$ : $\frac{d^{2} y}{d t^{2}}-5 \frac{d y}{d t}+6 y=0$ $y(0)=1, \quad y^{\prime}(0)=2$

Hello. So the question is taken formula place transformation. And uh we have to find the value of why using the laplace transformation in the given equation so that you and differential equation is little Y over the square plus two. The value was beauty Plus five of Y is equal to zero. And initial conditions are why yet is equal to zero is equal to two and second. This why Climb zero is equal to for okay taking the laplace transformation of on both sides. So we get s square foot office ffs is the fourier laplace transformation of this way minus Y. S way of zero minus y prime of zero plus two. As a full face minus two. By way of video plus five. Full fast Is equal to zero. Yeah as square. Therefore fast -Y is zero. Two ways -4 Plus two. S a full face minus four Plus five or 4. Face. I'm checking this will be yes that is fine. So so that will be taking a fo for storm comin. Yeah And that will be as square plus two. West minus five. That will be equal to To West Plus eight Into a four fish. So from here I for first will be Yeah. Yeah. To S-plus eight divided by a square Plus. to waste -5. So we can write it as to waste plus eight As -1 whole square plus two square and numerator. We can write details. M floor first two Into S -1 divided by S -1 Whole Square plus two square plus -2. 2. 10 divided by as -1 sq plus two square film. So I focused will be equal to to into the city because of duty plus five he to the T. Sign off duty. So from that is the uh sorry. So that will be very high value by taking that lap. Listen walls so that will be lipless transformation of why taking that lap. Listen. Also right hand side. We get the given solution of the equation. So hope this clears your doubt and thank you.

Yeah. Hello. So the question is taken from differential equation and the laplace transformation. And we have to solve this differential equation and the differential equation is. Mhm. Do you square Y over duty square plus? To divide over DT plus five by Is equal to 0? And initial conditions are by that is equal to zero is according to And why? Why am zero Is equal to four. Okay so we can solve the situation by using the laplace transformation. So taking a leftist transformation on both side we get S square efforts were emphasised lipless transformation of this wife minus S. Y. Zero minus why primed you plus two. S address -2. plus five. Have full fast is equal to zero. Okay so taking a first home on the left hand side of the time. On the right hand side we get a square Plus to west plus five into the office. That is equal to why is he always to the west plus four Minour going to do for my plus four. So from here the value of F. S. Will be to waste plus eight divided by a square plus to waste plus four plus one. Okay so that town become so let me write it here. F. S. Is equal to two. Yeah. As plus eight divided by S plus one old square plus two square. So we can I guess To into as plus one bless six. And this storm is divided by S plus one whole square plus two square. And this is also divided by X plus one square plus two square. Yeah. F. Office is equal to. Why? Why is the laplace transformation of this? So F. S. Is the laplace transformation of why? So why will be equal to to into exponential minus T. Because of two. Yeah plus three exponential minus T. Sign off duty. Which is the required solution of the situation. So who disclose your doubt and thank you?

All right. So our problem here is wise old prime plus two. I promptly fired White zero. Were given that Y zero. It was too with my prime zero, negative one started the applause transform the S squared why uh minus? It's going to be s times Y zero which means to me to us and it's going to be minus why privacy is negative one plus one. Yeah, plus two. As why are we going to be minus two times Y zero in minus for just five classrooms from why eagles here. Alright, separate them out. S square plus two S plus five, remove all the other terms to the right hand side. This can be to S minus one plus four, so plus three. Not to get the questions were alone will do pollsters from Y to go to to S Plus three all over S squared plus two S plus five. Okay. And now from here it kind of looks like we can work this into a form that we recognize and try to make this into the past transform of the exponential function as a senior sort of function. So first we're on now here around two F plus three all over. And then if you look here, you see the S square plus to us, you know, that's the beginning to the function as plus one squared, which we know will come out to s squared plus two S plus one. But we need to plus five. We're going to add another four. Now if you carry on. But the bombs we're gonna have to outside of S plus one. You need that up top for this for this form we're going for. Uh And then we need to get a plus three but it's going to get us to s plus to add another one. Here There you have s. Plus one squared plus two square. So I'm gonna write this two squared because in a loss transform for next bunch of function times as high as soil function. Um This term here is has to be a square. So now we're gonna separate this out. We'll get that too. Times S-plus one all over. S plus one squared plus two squared plus one over S plus one squared plus to start. All right? But here on the on this side here we know that these terms need to match. So rewrite this to S plus one over S plus one squared plus two squared Plus we'll take out 1/2 to make that equal. Well we have to up here S plus one squared plus two squared. So now we can solve especially why is he going to the applause transform of took two out. So I forgot uh S. Plus one over S. Plus one squared. Kind of messed it up. Yeah. Yes plus one squared plus two squared. Uh Plus one half. Universal boss transformed of two over S plus one squared plus two straight. Uh What we know from Y. Equals to eat a negative T. Uh co sign of two T. Plus one half E. To the negative T. Sign of two T. That is the answer to that bro.

In the problem we have Viral Illness -4. Mhm minus Plus five. White at Equal zero. Why have zero is equal to one? And why does zero is equal to two? Every day's a Squire. Lap plus white minus yes. Why is 0-? Why does zero minus forward into earth? Annapolis y minus Y zero plus five Last last. My that equals zero. So it is is required. Yeah plus Y- is -2 minus four. Is that plus one? Plus 4? Plus five. Hapless Way equals zero. So lap splice hawaii indu is a Squire minus forest Plus five. That equals S -2. Or lap list will become is minus to a bomb. Yes -2 whole squared plus one. So we have Y. F. D. Equal that blessing was all Is -2 upon is -2 Whole Square Plus one. So this is recognized you. The power minus sorry to the power to t costing. So we have we left becomes you have the power to D costing as our answer.


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