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E Ifz = x + iy.then we define Re(z) = x and Im(z) =y[2 points] Find =6 3i|2. [2 points] Find [2 pimts] Ifz =x+iy, find Re 22[2 pimts] Iz = x+iy, lind Im 22...

Question

E Ifz = x + iy.then we define Re(z) = x and Im(z) =y[2 points] Find =6 3i|2. [2 points] Find [2 pimts] Ifz =x+iy, find Re 22[2 pimts] Iz = x+iy, lind Im 22

E Ifz = x + iy.then we define Re(z) = x and Im(z) =y [2 points] Find =6 3i| 2. [2 points] Find [2 pimts] Ifz =x+iy, find Re 22 [2 pimts] Iz = x+iy, lind Im 22



Answers

$$\left\{\begin{aligned} y &=-2 e^{x} \\ 3 x-y &=2 \end{aligned}\right.$$ a. (-2,0) b. (0,-2) c. (0,-3) d. (-1,-5)

We need to solve law two plus X is equal to 2.5. To solve this, we take exponential on both decide so it is exponential. Log two plus X is equal to exponential toe point file. Now, using the inverse property we get two plus X is equal to exponential 2.5 hence X is equal to exponential 2.5 minus two.

If we had the equation three times E to the X plus two equals 75. And for the first one who wanted to test is X equals negative two plus. Either the 25th solution, we would substitute um negative two plus E. To the 25th for X. And ask ourselves is this true? Well I got a two plus two is zero. And so we have three E to the um you do the 25th equals 75. And that would mean E to the E to the 25th equals 25. And this is not true. So not a solution for B. We're testing what if X is equal to negative two plus the natural log of 25. So we would substitute that in for X three. E to the negative two plus natural log of 25 plus two equals 75 divide both sides by three. And combine like terms we have E to the natural log of 25 equals 25 exponential base E An actual log are in verses, so 25 equals 25 this is a solution and finally about 1.219 And so we would do three E to the 1.219 plus two equals 75. So you want to use a calculator here and if on your calculator three times E to the 1.219 plus two is about 75. So this is a solution as well. Yes.

Okay, So in this problem, we're told to compute the kernel of this function here or of this operator, this linear operator here, remember, the colonel is just when l Y is equal to zero. So l y first, this is going to be the squared plus two D minus 15. Why is equal to zero? So then we get d squared. That's right. This is just going to be D squared. Why? Minus two D? Why? Minus 15. Why? No, Why is equal to zero? Um, we writing that We can just say that's gonna be why Double prime minus two. Why Prime minus 15. Why is equal to zero. So the hint tells us we can look for two solutions of the form. Why is equal to e to the R. X here? So if we use that, remember why prime is equal to Ari to the my primacy. That's who are me to the Rx. Why? Double prime is equal to r squared B to the r. X. So, um, we have, um r squared each of the Rx minus two R e to the r X minus 15. He to the Rx is equal to zero so factor this factor out. Although not need to the Rx you have eaten the Rx times R squared minus two are minus 15 is equal to zero. Since either the Rx can never be equal to zero, we're going to set our squared minus two AR minus 13 equals zero. So you have R squared minus two. AR minus 15 is equal to zero. But we need to solve for R. Um, so this can actually factor. So we have our minus five r plus three here deals here. So remember, we just we take the factors of 15 which is, like, 135 and, um 135 and 15. Right. And then we look at two of them that have a difference of two, which happened be five and three. And then, since this is a negative, the bigger one, it goes with a negative. So are two solutions are gonna be when r is equal to five and our is you go to three here. So now, uh, are general solution is going to be, um so we have Why is equal to or write one is equal to e to the five x and then why to is going to be equal to eat to the sorry, actually be negative. Three negative three x. So our general solution is gonna be the two added together. So it colonel of L. There's going to be all the functions. See one e to the five X plus C two e to the negative three x where C one and C two are real numbers. Okay, and then that is our solution here.

In the execution we have to find the solution of the given differential equations of first of all, I'm writing the given equation that is Here to the power -2 wrote checks upon wrote checks minus y upon the road, checks into the X by device equals to one from here. I'm going to find the value of divide by dx So divide by dx is equal to the scene, you to depart minus, who takes upon the cortex minus why by pro text. Further I can say that this can be written as divide by dx plus, this is this will come here so this will be plus y upon route tex is equals to eat to depart minus two types of wrote checks upon wrote checks. Now compare this with general differential equation and we get these equals two, one by raw text and juries Into the par -2 who takes upon through text. Now integrating factor is given by that is if is equals two to the power integration three D X. So this will be equal to the to the power integration one by route tex B X. So this will becomes out to be into the power to rule text. This is our integrating factor. Now the solution is you can buy that is Y into integrating factor that is it to the powerful tax is equals to integration to into Italy for minus total tax upon tax into integrating factor that it into into the power to rule text the X. So this will comes out to be right into it to the power total tax equals to disease two types of root X plus C. So this is the solution of the given general equation and for that our option sees the correct answer. Thank you.


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