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In Exercises 7 and $8,$ make a change of variable that decouples the equation $\mathbf{x}^{\prime}=A \mathbf{x}$ . Write the equation $\mathbf{x}(t)=P \mathbf{y}(t)$ and show the calculation that leads to the uncoupled system $\mathbf{y}^{\prime}=D \mathbf{y},$ specifying $P$ and $D .$ $A$ as in Exercise 6

the uncoupled system is $\mathbf{y}^{\prime}=D \mathbf{y},$ where $D=\left[\begin{array}{cc}{-2} & {0} \\ {0} & {-1}\end{array}\right]$

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 7

Applications to Differential Equations

Vectors

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in this question. We want to decouple this early e using some sort of transformation where you define as this. So the first thing that we're gonna use is we're gonna use the alien vectors as a change of basis. So in particular, we're gonna let x b he why? And he is a good 1123 Why, so we have from our early E at X one crime your ex one minus y two x two an extra two crime musically to three x one months four x two And for my change of variables, we have x one is you two. Why one plus boy too? Why shoot and x two is eager to y one plus by three. Why, too? So I'm just reading it here, Andi to what we? What we have to do now is we have to substitute these two into these equations. So when we substitute disinterested equation, we had to take rerouted off. These say it's why one time that's like too wide to prime. He's even to write. One plus one is two right shoes my sweater to why want mice west six wide too. So this is just simple substitution into this equation, which is negative. Why one miles y for y two? And for our next question we have Why one crime plus by three. Why? To prime easy to you three. Why one plus by six. Why too, my school. Why one mice by 12 while shooting. So this is negative. Why one? My 1st 6 Why too? Okay, so have why one prime plus two y two. Crime is equal to native Why I want my spot. A why? And have why was frying plus by three Why to plan is equal to native y want my stress six wide too So de couple of these we just subtracted. Just sold for y two and by three. But why Toumai won't say this is attacking him We get why to prime is equal to negative six plus four gives you need to you too. Why don't you? And you have some for why one you can have? Why one prime He's executed. Why? One must work for y two mice White tube kind of needed too wide too. Budgetnegative wire One must work well. Why two plus by y two We just which is just going to wire one. And so you have decoupled your body

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