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Consider the following linear system: dx =X+y-Z dt dy =x+2y-z dt dz =y dta Find the characteristic equation of the system b. Find all the eigenvalues of the system ...

Question

Consider the following linear system: dx =X+y-Z dt dy =x+2y-z dt dz =y dta Find the characteristic equation of the system b. Find all the eigenvalues of the system C Find all the fundamental solutions of the system d. Find the general solution of the system

Consider the following linear system: dx =X+y-Z dt dy =x+2y-z dt dz =y dt a Find the characteristic equation of the system b. Find all the eigenvalues of the system C Find all the fundamental solutions of the system d. Find the general solution of the system



Answers

Use the variation-of-parameters method to determine a particular solution to the nonhomogeneous linear system $\mathbf{x}^{\prime}=A \mathbf{x}+\mathbf{b}$. Also find the general solution to the system. $$ A=\left[\begin{array}{rrr} 2 & -4 & 3 \\ -9 & -3 & -9 \\ 4 & 4 & 3 \end{array}\right], \mathbf{b}=\left[\begin{array}{c} e^{6 t} \\ 1 \\ 0 \end{array}\right] $$ $$ \text { [Hint: The eigenvalues of } A \text { are } \lambda=6,-3,-1 .] $$

Try to solve these to such problems with the separation of variables and the amount of integrating factors. So first do separation What variables with the 1st 1. And so this is going to give us dy dx Secret to -3 away. So then I have dy over Why is equal to -3 DX integrating both sides here I have the Helena wife is equal to You get three x. Let's see taking a to both sides here I get why is equal to C. E. To the negative three X against that state. First part by suppression of variables. Next this is my p sequel to three and so then have you is equal to to the In a group of three. She goes here to three X. So now I have the already axe of new times, why somebody to three X. Y is equal to zero times 8 to 3 X zero. So integrating both sides here, constant is equal to Into the three x. Y. So not dividing by into three X. Temple sites. Think of why is equal to see into the -3 X. Just the same. Okay. Second problem again, when you do separation of variables first I got T. Y. DT equals to white. Therefore dy over why Is equal to two DT integrating both sides. I get Elena like equals two T plus you. Okay take a nap both sides. I get why is equal to E. To the two T plus C. Here. She calls cE to the to T. That's my separation of variables here. Let's look at the other side now or the method by integrating method of integrating factors. So P is equal to -2. Therefore it is equal to here to the Integral of -2. We'll see you to the -2. X. Sliding that into that. So that's D. Or detox. Let's eat a -2 x. times y. And that's equal to zero Cereal Toast to zero. So then I integrate both sides. Get a constant is equal to It's a -2 x. That's why Divided by eating. They have 2x. to both sides. That ends up flipping that sign there we get like we'll see E. Two the two X. There should be cheers starting with all repair. There should be a team than to hear T. T. T. And we're done.

Okay, so then this problem were given this system of differential equations. So for part A is asking us to find the line where their infinite number of equilibrium remember, we find the equilibrium by sending the right hand side of this equation equal to zero. So that is gonna be to x minus Y equals zero and two x plus y equals zero. And as you can see here, these are both just multiples of each other. So, uh, if we roll reduce, we'll get just the system two x plus. Why equals zero, Right. So then we get the line y equals negative to X. So this is the line of our equal. Everyone points here. Now, let's graph the no clients for, uh, for part B to find the face plain. Okay, um, so to graft the XL claim that's the equation. Negative to x minus Y equals zero. Which leads us to the equation. Just y equals negative to X. Then if you were to graphic b y, no client, that's two x plus y equals zero, which again gives us why is equal to negative to X. So our face spot for this Okay, both the external climb and the wine. Oh, Klein are the same here. So let's plug in the test point. Say, you know, 10 re plug in 10 into the, uh Why? For the X on it. So too negative. Two times one minus one time zero sequel to negative, too, which is less than zero. So therefore, huh? Here they will be going to the left. Answers We're going to the left on this region and then to the right in this region. Okay, Now, for the next one. So two times one minus or sorry. Plus one time zero is equal to two, which is greater than zero. So things will be going, huh? Oh, okay. Oh, uh oh. Um, and then on the other side will be going down. Um, what this graph will look like essentially, is that solutions will be going How down like this. Let me think so. Combining them so our face thought will look like points. I go like this, huh? There we go. And they were done

In the problem we have the X upon D. D. Plus the way upon DT plus to Y. That is equal to scientists. Now the second equation is dx upon D. T. Plus the way upon DT -6 -1 That equals to zero. So we have dx plus D. Plus two Y. That equals scientists 1st equation and then Day -1 X Plus The -1 Y. That equals zero. This is 2nd illusion. So further we have to reply The U -1 into fast aggression minus D plus two into second equation. Therefore we have Day in two D -1. X plus Do U -1 and today plus two Y. That he calls D -1 sci fi. Now we have the other part that is deep plus two In two D -1 x Plus D -1 and two D plus two Y. That is equal to zero. Now we have to subtract these parts. So subtraction here. This podcasters each other. So further we have the Squire minus d minus the esquire Plus two D -D minus two X. That gives us cost e minus scientists. So this this is equal to the Squire minus t minus t squired -D-plus two X. So this cancer is this part and we have cost minus scientists. Finally it is minus stoolie plus two X. That is equal to costing minus Sandy so far that it is D -1 X. That is equal to scientific minus go straight up onto further. We have the general solution to the difference of liquidation that has given us X. That is equal to C. U. To the power t minus half scientific. Now for that we have to find for why. So yeah now we have to find for why. So to subtract the second from first equation. So it is 1 -2 gives us D- D -1. X plus D plus two minus day minus one. Why? Which is equal to 70? So father has become d minus D plus one. X plus D plus to minus D plus one. This cancels this part these kinds of this part there four days equal to 70. Now we have X Plus three Y. Which is equal to 70. So further three Y. is equal to 70 -C. E. The party plus half scientist. Because you have to minus x. Two X. S mm This is X which is saved the party -170. So here this becomes this now further why is equal to one up on three society minus And upon three ce party Plus one upon 6. Scientific. So finally why is equal to went up onto 70 -1? Up on 3? Say it party. So we have this as the answer to the problem


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