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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. $$A=\left[\begin{array}{ll}{-3} & {2} \\ {-1} & {-1}\end{array}\right]$$

$\mathbf{x}(t)=c_{1} \mathbf{v}_{1} e^{(-2+i) t}+c_{2} \mathbf{v}_{2} e^{(-2-i) t}$

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 7

Applications to Differential Equations

Vectors

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Okay, so we're given to me a goes my human. It's 111 arrest solve assist in X equals. It's probably because first we have to find value in values and collectors of a one of the idea though salted value that we have with you Geico to my team. Plus I already you see that the the real part of the even though he negative it implies that you will have spirals in your feet portrait and it take a note been inward to the second part of the question that we can automatically solve right now Analysis pointing the general solution. So the first point are you vector two bodies keep I is one minus I want have you can construct our submission the one time e V as in Victor in my eye Im ki you. The other solution is just because you get a solution This one just minus que mind If I v one you hear and then no, you know we can write Need the money What I times t using orders formula we get I need the money. Cuchi co sign. He was fine. This is the word for it. And then we can just spoke this into imaginary in real part. First, we'll expand this out. We get U minus Q t co sider He I signed he I defector the one defector I infected. You know, when you do that you get Was he minus I Co sci fi. What? I find p and my ice We're fine tea. The second rotation A coast I he off site then when you split into the imaginary in real part, you got so sorry people sign t over co sci fi and you have the legend report 70 my co sign T psyche. Really? We use the one that of this port years because you get solution that air con you There's, you know, we have a real and imaginary part. I mean, our solution Just the one times Mike cuchi coastline api flying you, he and you, you can like see, I'm signed he my coast he and you know before, Like I said, the this is the spirals because we have complex like in value and they get a spiral towards or you because the idea like the real part, I think like value value, negative

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