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Find the general solution to the system of linear differential equations The independent variable is t. All of the eigenvalues and two of the eigenvectors are provi...

Question

Find the general solution to the system of linear differential equations The independent variable is t. All of the eigenvalues and two of the eigenvectors are provided.x; = X1 + 2xz + X3 x2 = 6x1 - Xz x3 = -X1 - 2xz ~ X311 = 0,12 = 3,13 = -4K1 = 13Kz

Find the general solution to the system of linear differential equations The independent variable is t. All of the eigenvalues and two of the eigenvectors are provided. x; = X1 + 2xz + X3 x2 = 6x1 - Xz x3 = -X1 - 2xz ~ X3 11 = 0, 12 = 3, 13 = -4 K1 = 13 Kz



Answers

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

Okay, so we have this differential equation here and we need to find solutions of the form A Y of X is equal to eat of the Rx. So first we need to find why prime. So why Prime X is equal to R E to the R X, Then why double prime of X is equal to our square either the Rx and finally, why Triple prime of X is equal to r cubed e to the R. X. So plugging all of these into this equation here we have, ah y triple prime is our Q E to the R X plus three r squared E to the Rx minus four R e to the r X minus 12 e to the R X is equal to zero. We can factor out and eat of the Rx from all of the terms. So we have either The Rx seven R cubed plus three R squared minus four are minus 12 is equal to zero. So now we, um, since we know that either the Rx can never be equal to zero, then we just need to set this part equal to zero. So that's our cue. Plus three R squared minus four R minus 12 is equal to zero. Um, now, to factor this, we can use the rational luthier. Um um, since, um, this condition here is one we only need to look at the factors of 12 here. So the factors or 12 or one, You know, plus or minus one close to minus to plus or minus three plus or minus +46 And then finally just, uh, a six or plus or minus 12. Right. So let's use synthetic division to try to, um, see, which of these factors are going to be a factor or Yeah, read. So let's put a co efficiency or 13 negative four Negative 12. And let's just start with, um, I'm gonna guess one for now. Okay. So one always just carries down here, then we'll take one and one. So that becomes one. You had it. 44 here become 00 And the negatives. Well, this last number is not zero. So one is not going to be a route. So let's start over. Okay, let's try a negative one. Try negative one. Right. Let's go through this. So native three plus negative one is gonna be negative, too. A native one times negative two is positive to negative two and this becomes to hear negative 10 again, not zero. So let's let's move onto the next one. Let's try positive too. Okay, so two here. So to want two times one here. So then 52 times five is 10. Then this becomes I'm sorry. It's become 62 times sixties deposited 12 0 Great. So now we found one of our roots is too, so we can factor that into our minus two. And then r squared plus five are plus six is equal to zero again. We can factor this one here. Six is going to be equal to two times three huts and then two plus three is five. So we can turned this into R plus two and then our plus three here groups R plus three is equal to zero. So our roots we're going to be to native to a negative three. So our three equations are going to be e to the group's e to the two x needs of the negative two x and eats the negative three x

For this equation, we need to find solutions of the form wise equal to eat the Rx. So next we need to find why Prime, which is gonna be our either the Rx by a double time, which is equal to R squared either the Rx. And then why Cube, which is equal to our cube either the Rx. So now we substitute these into our differential equation here. So we get our cube. You know, there are X plus three R squared. Either the r X minus four are either the Rx minus 12 e to the R. X, and that's equal to zero next week. In fact, out in either the Rx from all four terms here. So we get either the Rx and then what's left overs are a cube plus three R squared minus four. R minus 12 is equal to zero. Since either the Rx can never be equal to zero, we need to set this part of the equation included. Zero. That's our cue. Plus tree R squared minus four. R minus 12 is equal to zero. Since this, um, constant coefficient here. Is it with one by the rational root beer? Um, we only need to look at the factors of 12. So includes plus or minus one plus or minus 12 plus or minus to plus or minus six plus or minus three and poster minus four. Right. So we're going to go and try to find the roots of this by using synthetic division. So 13 negative four and negative 12. So we contest which of these are going to be roots. Okay, so I want to start with one to see if one is a root of this equation. So one. So I bring down the 1st 1 then one times one is one. Then I add those to you. I get four. Um, then one times four is four. That becomes 00 and negative. 12. Since I don't have a zero here, one is not a route. So I'm gonna start over, so erase those. So now I'm gonna try a negative one. Okay, so let's try. Negative one. One times. Negative One is negative. One that becomes three plus negative. One is to hear one times negative. Two is negative. Two is becomes negative. Six, which comes positive. 12. Um, sorry. Um, this becomes positive. Six and again it's not. Ah, so this is not a route as well. Okay, um, so now let's try to okay, Race nous. So let's try to here so to I'm gonna try to times one is two. They gives five here. 10 um, so 10 minus four is six than two times six is 12 0 So two is a root of this equation here. So if we factor that we get our minus two and then are left over coefficients and we are square plus five r plus six is equal to zero. So now we can factor this further. The factors of six at up to five are two and three, so that's gonna factor into our plus two. And r plus three is equal to zero. Therefore, our roots are gonna be r equals too negative too. And negative three. So we're gonna hav e to the two x e to the negative two x and e to the negative three x as are linearly independent solutions Now our general solution. Why Sea of X is going to be equal to C one e to the two x plus c two e to the negative two x plus c three e to the negative three x

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


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