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Sec 3.9: Problem 3Prevlous ProblemProblem LIstNext Problempolnt) Consider the nonhomogeneous sysiem"=[ %+[2]Write - fundamental matnx for the assoc ated homoge...

Question

Sec 3.9: Problem 3Prevlous ProblemProblem LIstNext Problempolnt) Consider the nonhomogeneous sysiem"=[ %+[2]Write - fundamental matnx for the assoc ated homogeneous Systemreip (forulas) help (matrices)Compute the Inversehelp (tonulas nelp (matricos}Muitiply by the nonhomogenelty and Integratefx' j at = Do not Include and € In your answers Glve tha solutlan t0 {he systemnelp (ommulas) help (matricas)(tormulas}help (matrices

sec 3.9: Problem 3 Prevlous Problem Problem LIst Next Problem polnt) Consider the nonhomogeneous sysiem "=[ %+[2] Write - fundamental matnx for the assoc ated homogeneous System reip (forulas) help (matrices) Compute the Inverse help (tonulas nelp (matricos} Muitiply by the nonhomogenelty and Integrate fx' j at = Do not Include and € In your answers Glve tha solutlan t0 {he system nelp (ommulas) help (matricas) (tormulas}help (matrices



Answers

28. Let $\mathbf{X}(t)$ be a fundamental matrix for the system
$\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x} .$ Show that $\mathbf{x}(t)=\mathbf{X}(t)\mathbf{X}^{-1}\left(t_{0}\right) \mathbf{x}_{0}$ is the solu-
tion to the initial value problem $\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}, \mathbf{x}\left(t_{0}\right)=\mathbf{x}_{0.}$

In this video, we're gonna go through the answer to question number 19 from chapter 9.5. So we're asked to find the for a fundamental matrix of wth e differential equation X prime is equal to a X where a is given by this two by two matrix. So first gotta find the item values, they're gonna find their respective, and I come back to us and inform e fundamental waitress excites find aiken bodies. We need Thio, uh, find the values off which the determinant off a minus. Ah, I people's ear. And so we can ride it that as a servant off minus one. My, uh, one aids one minus, huh? On that seven of this is just as a top left element times by the bottom, right element minus bomb lifetimes Top. Right. So this is gonna be just explaining the buckets. B r squared minus one minus. AIDS just minus nine. So we're not equals zero, then. Ah, it's gonna be equal to plus or minus the square of nine, which is three. Okay, so there are two aiken values, puts a minus three. It's not gonna find it effective. So we've gotta look for a factor. You such that? A minus that I value times the identity matrix times by you. That's what you want for the first I can value is equal to zero. Okay, so what's that? Matrix world? A minus three. I gonna be minus four one aids minus two times by. You want deep zero. So this means that the first element of you one if that is just equal to one, then the second element is gonna be equal to four. And you can show that from either the top rope or bomber at both Make the same equation. Okay, for the next one, find a three i times by. Let's call it u two. I was there, So that's gonna bay two one day. Well, because by some back to you to zero that satisfy this you two, you could be equal. Thio. Well, if the first value one, then second gonna be minus two. Okay, then our matrix is gonna be for the first column. Is gonna be Theo first. I value time first better. Yes. I said eat of our first line by your times. Two times. By that, I could better So yeah, the 1st 1 This is gonna be e to the three tea. That's good. Before eating three team top right is gonna be well, tough, right? Actually, one time by eats the minus treaty. Okay. On bottom, Right. It's gonna be minus two e to the minus three t. That's how fundamental matrix.

In this video, we're gonna go through the answer to question number 11 from chapter 9.5. So asked to find the general solution of the system. Ex Prime is is equal to X for the Matrix given by a Yeah. So first, let's find the Eiken values me to find the corresponding island backers. And then we can work out what the German solutions off the system of equations is. Let's find the item values. Let's find the determinant off the matrix. A minus times insta matrix. I says the matrix minus one minus R 3/4 minus five three minutes Are That's gonna be equal. Thio one course. Uh, times by a minus three. I've just taken the AA minus one for the first bucket into the second bracket. Minus minus five times to be over four. That's 15. It was all so that's gonna be equal. T squared minus three aa plus, uh, minus two are minus three plus 15 over four minus three is minus 12 hour before course 50. No, before is three or four. Okay, then we can fact arise this two aa minus, huh? And, uh, minus to be over too. So If we said that equals zero, then yeah, good values are is equal to half on. Dhe obviously goes with three of it. There are values next door when it's fine that I come back to us, says Rita. Let's first find the island back to Christ. Wanted to involve you. Uh, so, Amos half I it's minus three of it too. Three or four minus five. Three minus 1/2 is fight over to you because by you won't what? Zero. Okay, so if we let you want the currency, you won't be X y that first row tells us back. Modest three over two. Thanks. Close three over four. Why equals zero? So therefore, why is equal to? Well, goodbye, all by three. Then it's gonna be y is equal to for over two ex, which is just two thanks. Effects equal to one that to that Sorry that y is equal to two. Next up the Eiken vector for I can value as he was three over two. So minus one minus three over two. It's minus five every day. Three of a four minus five than three minus three. Opportunity is just to be over too okay. Again letting you won. I have components. Sex. And why then, minus five over two times X close, Clear before why equals zero. Therefore, why will be equal to okay? What's gonna be five over too? X times by for over three. So what's this? It's gonna be 20. Ah, let's go to 10 over three times X. Okay, so why is 10 over three of X? So relax. Be re then why is gonna be tough? So that's, uh, second Aiken vector. Okay, so now those to you I am vectors linearly independent. So we know that the jealously shin gonna be X dysfunctional T It's gonna be okay. A constant c one size by eats the half t hard being our first I blow you times by the item back Their course one is that I can value, which was one two plus say a second constant times e to the second. I can value t times by the second night connector, which was three. That's not a solution

So for this problem we have that the memory, so we'll have the M one plus M two. So these are memories. So the memory of one plus the memory of two has to give us 252 136 mega bikes. And then if one program, so we'll say M1, the first program um or is equal to 20 more than twice the memory of the other, so to M2 plus 20. So if we subtract over the two M two, we get that are coefficients are one -2, and this is equal to 20. But when we do this, we see that the first one's memory needs to be 164 MB, and the second one needs to be 72 MB. And this makes sense because if we add them up together, we end up getting that. This is going to be Um 236, just like we expected. And then we can also convert this into gigabytes if we wanted to. But this will be the final answer in megabytes. Yeah.

In this problem. We need to find all my cases off home. Easy do. Yeah. Zero c. So that is where it was. Who I No, it is given. That is ways. Bull's eye. Therefore we can ride. This s is it'll see Miley by a zero c was too. I like his ideas. Humor. It's one real zero. Are you creating the corresponding down on these equal matrices? Get a square. It was one C square is close to one on ISS where less easy was zero the board a square. But these guys, what's to one Which means that a and C can be plus minus one. And it's where less NC is close to a deal named this as immigration one. We can also write this as a mullet like a plus c was Zebo or a plus c reports to video and is Nordic where zero Andi is equal to minus C named this as a creation toe, not from immigration Morning toe We get that if equals to one see the close to one and is it close to minus? One season was one then the possible entries is all the home is It is is equals to a zero C Gonna be buzz 110 minus one and was second. It will be minus one minus one and Tzeitel one, Therefore, that's a solution.


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