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Solve ux (2ydy0 by introducinginlegraling facLor Lo Hlake the equation exact...

Question

Solve ux (2ydy0 by introducinginlegraling facLor Lo Hlake the equation exact

Solve ux (2y dy 0 by introducing inlegraling facLor Lo Hlake the equation exact



Answers

Solve the differential equation.

$ y'' + 2y = 0 $

In this problem, we have to solve the differential equation. Why? Double crime minus y prime minus six. Why is he called to zero so forthis kind of differential equation? We can a specific characteristic in creation. So the characteristic decoration it's gonna look like the square minus T minus six. Is it called a zero? Now let's try to solve this equation. I can see that there is a nice factory ization. This can be factored as t plus two dines. T minus three is a call to zero so that in place either tease a call to negative to Ortiz is a call to positive. Three. Certain The fundamental solution are going to be Ito the poor. Negative two times six on each of the power, three times six. So then, general solution, it's going to look like Why is he going to see one? For some constant? It'll depart negative to x plus C two you to the poor three x So that's going to be might general solution

In this problem, we have to solve the differential equation. Why Double prime minus six y prime plus nine y is equal to zero so forthis kind of differential equation. We can always associative characteristic a creation, which is going to be D squared minus six times D plus nine years ago to zero. Now this implies D minus three. Whole squared is equal to zero thirties. They called to three. In this case, we see that we have only one solution. So the fundamental solution it's going to be easy to the power three x and x Times E to depart three extra. Remember anybody on the same route? Then the fundamental solutions becomes X times E to the poor that route time. Six. So then that general solution it's going to be some constant c. One times e to depart three Ex Placido times X times It'll be poor three X

Okay. So to determine the general solution of this different situation, we first going to find the, uh, auxiliary equation. So this is going to be equal to R squared minus R minus two. And we're going to set that equal to zero, so we can now find the reach of this so this can be factored here. So it's going to be equal to AR minus two, and then our my are plus one is equal to zero. This gives us two routes of, uh, negative one and two. So our general equation Why of X is going to be able to see one e to the negative x plus seat two plus C two e to the two x.

All right. So, in. To find the general solution of this equation, let's first turn this into differential operator form. So that becomes D squared. Um, and then B squared minus D minus two of why is equal to zero. So then now are corresponding. Auxiliary equation P of our is equal to r squared minus R minus two on. We need to say that able to zero. So if we want to attempt of after this, we take a look at the factors that to which are just one into, um, and look at the difference. So one and two is going to be negative one. So that matches this here. So then we can talk to this into our minus two and r plus one single zero. So our roots are negative one and two. So our general solution Why of X is going to be able to see one e to the negative X plus C two, e two, the two X right. So this is our general solution here.


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