Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

In Exercises $9-18$ , construct the general solution of $\mathbf{x}^{\prime}=A \mathbf{x}$ involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories. $$A=\left[\begin{array}{ll}{-2} & {1} \\ {-8} & {2}\end{array}\right]$$

$\mathbf{x}(t)=c_{1}\left[\begin{array}{c}{\cos 2 t+\sin 2 t} \\ {4 \cos 2 t}\end{array}\right]+c_{2}\left[\begin{array}{c}{\sin 2 t-\cos 2 t} \\ {4 \sin 2 t}\end{array}\right]$

07:36

Ibrahima B.

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 7

Applications to Differential Equations

Vectors

Johns Hopkins University

Campbell University

Harvey Mudd College

University of Nottingham

Lectures

02:56

In mathematics, a vector (…

06:36

06:17

In Exercises $9-18$ , cons…

04:26

03:01

01:31

05:08

15:08

In Exercises $9-18,$ const…

03:37

In Exercises 21–36, find a…

01:32

Solve the differential equ…

02:26

01:53

so for us to find the solution to this differential pressure, remember, the first thing we're going to do is we want to write this in the form of our solution being so some constancy one times our first vector or Eigen vector and then e to the Lambda won t where Lambda One is our first, um, Aiken value. And then we just repeat. But just with tubes everywhere. So this is the form gonna end up wanting to get in, and they tell us our solution, they're gonna be complex. And then we could talk about how to turn them into the real ones after we get the complex solution. Right. So let's go ahead and write that down. Eso we first need for what are Eigen? Values are going to be. So we do a my slammed I, which is going to be negative two minus lambda One negative eight to minus lambda. And remember, if we take the determinant of this, that should be equal to zero. So we get negative two minus lambda tu minus lambda. And then I would be plus eight zero. And if we were to expand this out now, we would get? Um, actually, let's pull that negative out. So would be negative here. And then this is just gonna be the difference of squares. So this would be four minus lamb to square plus eight. And that's gonna be zero. And now we can just go ahead and distribute the negative. So that would be Lambda Square to B minus eight plus eight to the B plus. Or is he go to zero? And then we could go ahead and move everything over. So Lambda Square, it is going to be made before. And then we take the square root on each side. And so then that gives us Lambda is going to be able to plus or minus two I and actually from that not x lambda. And actually, from this here, we can already go ahead and figure out what the trajectory off these were going to be. Because if we have something in the form of a plus B, I so if a is zero so that if a zero, then we know we will have ellipsis of Lipsky's of round the origin. So this is going to be the trajectory of it. Since over here are a is zero. So that's the trajectory. But now let's go ahead and find one of our Eigen vectors of and go ahead and pick this up. And remember, we only really need to find one of the icon vectors since for the complex ones thes second Eigen vector is just going to be its complex conjugate. So if we were to go ahead and do lambda or no, not Lambda but a minus, uh, to I I this is going to give minus two minus two. I negative 81 to minus to I and we could go ahead and do like the road reduction to make sure that the bottom row is going to be zero. But since these are complex, we know it will be the case. So we could just be a little bit lazy. And I'm just going to use that last one there. Eso I'm going to rewrite this as negative eight x one and you can also use the first one. Doesn't really matter. But I just think the numbers down there, uh, well, solve for X one a little bit easier, since that's normally will be solved for okay, next to me said that he was zero. And now I'm gonna add X one over. So be eight x one. There's a two to minus two I next to and then we divide the eight over. So x one is going to be equal to So the one minus one I over four, uh, to. So when we come over here and create our Eigen vector, this is going to be so one minus I over four, uh, X two and then we have X two here. Then we could pull the excuse out so we'd be left with that. And remember, now we could multiply this by anything we want. So I'm just gonna multiply this by four. And so our first Eigen vector is going to be one minus I for all right. And so, actually, let's write this down here. So we have when Lambda is equal to two I we get B one to B one, minus I for and that means for our complex conjugate or second hike in vector is just gonna be one plus I or because we just make it the complex conjugation. So those are going to be our sectors now. And what we can do is go ahead and plug it into like, that equation. We had a top there, so are complex. Um, valued one is going to be X is equal to So it was C one and then be one. So one minus side for and then e to the one or not 12 i t. And then plus C two one plus I or and then eat negative to I teeth. And so this is going to be the complex solution. And then we can go ahead and get the rial valued solution. Um, by expanding this and then rewriting everything. But they actually give us a nice way in the book to do this. And they say that we can write this as so the X is equal to C one y one plus C two y two, where why one is equal to e riel of one of the Eigen vectors. So it doesn't matter which one we use, plus cosine Bt and then minus the imaginary portion of one of our Eigen vectors and the sign of Bt. And then all of this is multiplied by E to the 80 and then for y two we like, switch the signs and co signs we keep the vector the same And then the subtraction is going to become addition. You could get this just by, like, using the E to the I data is co signed data, plus I sine data. But I think doing it this way is a little bit, uh, cool quicker, because then we don't have to, like, think about, like, combinations of things. But you could do it that way as well. You'll get a similar answer. It might just be a little different due to, like, combinations and stuff, but it'll be the same thing in, like, the grand scheme of things. But let's go ahead and plug everything in, although so I'm just gonna use Lambda One and we want So in this case, slammed of one is going to be. And remember, this is supposed to be like a plus B I. So that's where our A's and the's air coming from. So that was just zero plus route to I, and now we can get our riel portion of the vector and imagining by breaking that up. So the real portion of this is just one or and then we add this to would be negative 10 Then we have either, So we could just come down here. So this is the rule sourcing. This is the real and this is the imaginary. So let's go ahead and plug those in. So the reels are going to be 14 and depending on if you started with a different Eigen vector, um, you would get something slightly different here, but again, it should all give you the same thing as long as you're consistent with everything else. And then the imaginary should be negative. 10 negative 10 for our B that's going to be not route to just to. So let's go ahead and replace all these bees here with twos. So to to to to And then a was supposed to be zero. So actually, this is just e 20 which that's just one. So we can kind of just ignore that on. Then we just need to multiply the stop one. I see one c one, c two, C two, and we can go ahead and, um, collect all of the collect all the coastlines together and the signs so that gives us X is going to be so let's see. Um, so this would be the cosign here along with this and doing that, we will end up with, um, are very I'll just add these together. I'll keep the coastlines and signs the same. So this is just gonna be C one. And so this is going to be host sign to t and then minus minus, actually, let's just go ahead and make that a plus there and there will be plus sign two teeth and then this would be just four co sign to teak and then plus to see, And then it would be so sign of two t and then minus co sign of two teeth and then down below it before sign of two teams. So this would be our solution with the rial Once, um, I guess you could try toe, make this in tow. One bigger one. But I'm just gonna leave it like this here and so again, you might have slightly different numbers, um, in front of them. But as long as you can, like, pull all those factors out, you should get exactly what we have here.

View More Answers From This Book

Find Another Textbook

In mathematics, a vector (from the Latin word "vehere" meaning &qu…

In mathematics, a vector (from the Latin "mover") is a geometric o…

In Exercises $9-18$ , construct the general solution of $\mathbf{x}^{\prime}…

In Exercises $9-18,$ construct the general solution of $\mathbf{x}^{\prime}=…

In Exercises 21–36, find an equation in standard form for the ellipse that s…

Solve the differential equation in Exercises $9-18$.$$\frac{d y}{d x…

Solve the differential equation in Exercises $9-18$.$$\sqrt{2 x y} \…

Solve the differential equation in Exercises $9-18$.$$2 \sqrt{x y} \…

02:34

Let $V$ be a vector space with a basis $\mathcal{B}=\left\{\mathbf{b}_{1}, \…

00:45

Exercises $9-14$ require techniques from Section $3.1 .$ Find the characteri…

01:46

Suppose $\mu$ is an eigenvalue of the $B$ in Exercise $15,$ and that $\mathb…

06:38

A certain experiment produces the data $(1,1.8),(2,2.7),$ $(3,3.4),(4,3.8),(…

01:07

01:03

Exercises $3-8$ refer to $\mathbb{P}_{2}$ with the inner product given by ev…

00:53

04:00

Let $y_{k}=k^{2}$ and $z_{k}=2 k|k|$ . Are the signals $\left\{y_{k}\right\}…

In Exercises 17 and $18, A$ is an $m \times n$ matrix and $b$ is in $\mathbb…

04:42

In Exercises 17 and 18 , $A$ is an $m \times n$ matrix. Mark each statement …

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.