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Find fundenkentul *tol solutian eclor of Lhe sYslem *AwhereIhe given matrix:find the solution that sutisfie $ tlic initial condition:~o-/3Hint:-6 is cigenvalucX()=#...

Question

Find fundenkentul *tol solutian eclor of Lhe sYslem *AwhereIhe given matrix:find the solution that sutisfie $ tlic initial condition:~o-/3Hint:-6 is cigenvalucX()=#ia+~18 X(F 43++~H X()= #14+1+4 X(IF #lil#lekIH X()= #"1 "14d None of the above_

Find fundenkentul *tol solutian eclor of Lhe sYslem * Awhere Ihe given matrix: find the solution that sutisfie $ tlic initial condition: ~o-/3 Hint:-6 is cigenvaluc X()= #ia+~18 X(F 43++~H X()= #14+1+4 X(IF #lil#lekIH X()= #"1 "14d None of the above_



Answers

First write the given system of differential equations in matrix form, and then use the ideas from this section to determine all solutions. $$\begin{aligned}&x_{1}^{\prime}=x_{1}+x_{2}-x_{3}, \quad x_{2}^{\prime}=x_{1}+x_{2}+x_{3}\\ &x_{3}^{\prime}=-x_{1}+x_{2}+x_{3}\end{aligned}$$

The aim of this exercise is the soul of the system of three differential equations which can be written as shown on the screen. The characteristic polynomial off the matrix A in this case is given by minus lambda minus two times and then the When is three this square I'm not listed on When we solved the characteristic equation that is this'll equals to zero. When we find that a has to Eigen values I am the one equals two minus two. Aunt um, the two equals 23 no, on American factory corresponding to them, the one is 1111 and notice that I'm the one had outbreak multiplicity of one. Therefore, all off the Eigen vectors corresponding to Lambda one are going to be scaler multiples off the one Andi. Since Lambda Two has anti fragmented publicity off to then we hope we can find to I am vectors corresponding to the Eigen value three, which are linearly independent. Andi, this can be done. So we have fit to equal to 100 That is one again victory corresponding to die in value three and also feet three equals 01 and minus four It's an Eigen vector corresponding to the Eigen. Value three on B three and B two are linearly independent. So Ah, now we conclude that a is Evan realizable aunt the lights of more. And that, if we said is to be the Matrix, whose first column is 111 whose second column is 100 Her stirred column is 01 on minus four. Then we have the s immersed in state is posted that they have no matrix minus 23 three zeros anywhere else. Now, from here we can write a US S times the times as immerse now going back to our system of differential equations. We had ex prime us Hey, eggs And if we write a uh huh s time, See, Time's as in virgin eggs and we multiply by IHS Immersed in the left we could is immersed stems ex prime equals the times is immersed x And so here we make a change of variable ive why he's equal to hiss inverse eggs. Then we have the following equation Why prime equals the Y That is why one and why? To equal to minus two on Dwight three praying, praying praying equals two minus 200030003 times. Why? One way to runway three? No, these are differential equations for which we know the solution. In fact, why one equals to see one times e to the minus two t Why two equals C two times e to the three. D on why three equals C three times heat with three t. Now to go back to our original, um, system of differential equation, we have that X is equal to s. Why therefore, she had been generated on green eggs, which is equal to x one x two x three is equal to the metrics. S 100 she wrote one minus four times. See one eat the minus two D See to eat three t and C 383 thing. Andi, When we make the product we obtained x one x two The next three have fixed following expressions. See one eat the minus two T plus C dude, eat the three D then t 18 to the minus two T plus C three mhm three D on DFO. Finally see 18 to the minus team minus four times C three. Eat three teams on. This is our solution off the system of differential equations

Okay so to start off this problem I'm going to go ahead and rewrite the right hand side. So the RDS is going to be E. To the R. Times E. To the negative to S. And this is well we can go ahead and move the head of the R. To the left hand side. So I've won over eat to the R. D. R. Equals to one over E. To to S. Diaz. Okay and so I'm gonna integrate both sides. So the left hand side it's going to integrate too negative E. To the negative are Mhm. And the right hand side it's gonna integrate to um negative one half. Okay. Eat a negative to S. Plus C. Yeah we can multiply by negative. So I have that. Eat the negative R. Equals to one half E. To the negative to S. Plus C. And we can also ray uh take the natural log of everything. So when we do that we have negative are equals two. The natural law. Yeah of the absolute value of one half each. The Negev to S plus E. And we can multiply by negative. So we have our equals to the natural log the apps of value of one half. Eat the knife to S. Plus C. Times -1. OK. And now we can plug in our initial value conditions which were as our of serial zero. So we have that zero equals the natural log The asset value 1/2. Eat to the zero power is just gonna be one. sorry 1. So this is going to be that We can multiply a negative one. Okay So we'll have zero equals to the natural log the absolute value of 1/2 plus c. Right? And then we can also get that C. needs to be 1/2 Since the natural log of one equals to zero. Okay. And so now we can plug this in back into our equation to get our final answer. So r. equals two. The natural the negative of the natural log, the absolute value of 1/2 mm to the negative to S My sorry, plus 1/2. And actually I'm just going to remove this absolute value signs since I didn't actually in a great for it, and that's gonna be my final answer.

So we have this one still gonna be a separation of variables. Right? So this is, uh, uh, under the differential equation. So we're gonna is a separable differential. Equations they were gonna separate received two separate the variables. This is gonna be 1/2 I plus e to the negative T j plus one over t plus one K. Right. And then you have DT year, and then you integrate both sides is gonna be our equals. What is the the French the integral off this one. Those t plus one here. Eso is gonna be okay. You have 3/2. Then this one is gonna be t plus one. You know, 1/2 plus one. That is the interval. You're gonna add one to it and then divide by, uh, that same thing. So when I add 1 to 1/2 is three over to that I'm divide by three over to Dallas is gonna be the same us or to playing by two or three. Right. And this is gonna be I than this is gonna be. Ah, I e too. T j. There's the integration of eater. Negative. T is the same as negativity to the negativity and employees. This one is gonna be natural. Log off T plus one and then key. Right. So you have this one. Mind you, this one is gonna cancel for you to have one right now are zero is K brain. So arms, You know, this is our of tea. So are zero means that we're a c t. I'm gonna put zero. So this is gonna be zero plus one, right to the power 3/2. I write, and this is gonna be he to the negative zero j. Then there's gonna be natural zero plus one K. Right. And the results I'm gonna equated to K, right? That is the initial condition. So I'm quitting this 12 k. There's gonna be a Pelosi here because it is a new indefinite, integral. And you're supposed to provide arbitrary, constant C, Right. So what is gonna happen? So this is this run Just one right here is one to the party to which is still one. So this is gonna be just one. I would just still one and the ease the negative zero. It still is either zero, which is also one so J and then, uh, this one right here is on Lin off one right national level one here is zero. Right. So this is in a B plus, see? And that is equal to care. Right? So it is C and C is gonna be negative. I plus J plus K right. That is gonna be our c. So we're gonna do this in different page. So in place of this, See Right here is it to see right here? I'm gonna put negative I plus J plus Kate. Right? Eso when I do that, when I might get in, So are of t is gonna be, um uh t plus one right T plus one to the voluntary over two. So there's gonna be t plus one to the power three over to I. And then So you have this one, and then the 2nd 1 is gonna b e t o negative. Either. The negative t j. And the 3rd 1 is natural natural log T plus one k and then our see this are See Here is this time this one. Right? So I'm gonna put it there. So when we put a debt there, what is happening? This is gonna be negative. I plus J plus Kate. Now we're gonna group like terms, right? Have I got jg any God, KK? You're a so gonna you're gonna group like terms here, So this is gonna be, uh, t plus one to the power 3/2 minus one. Right. And there's gonna have an eye, and then this isn't the, uh plus one minus. You did it negative. T j and then gonna have natural law t close one plus one, right? And then k here. And that is gonna be your r t as your fans finally.

The discussion. We have the difference because in our density it's a good like menace cap and having his Children this scenario zero they will tow my campus. Kick up artistic Significant and Steve 80 Deal Model P. This is my cab business, Jacob. This you don't kick out. So the a lot off the physical like a shake up and it will get up. Good ET sent a knitting on both the sides Figured out off the secret book Dangerous Rip Individual components. Digress Enough duty like Venice Indicators deep Jacob Is he able? Rookie? I kept man ist Jacob. Let's see. Corruptive. This is Jim Nationalism. No substituting the in the seven is an autopsy. Rose, who I got because he was just three K cup is equal to D. I have Manistee, Jacob. Let's see. So you have to use you cuz gonna go like up. Let's take your cap Putting back in Dominica's photography the single the Jacob Manistee. Jacob, that's too like this. Three. Jacob The physicality has to I get mine. STG gap This three kick up so I don't beat music for this. This is the for taking muscle yourself. Uh huh.


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