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(15 points_ Solve the following initial value problem: You can use Wolfram; Alpha to evaluate integrals, but must show solution steps.y +ry = ry y(0) = V2...

Question

(15 points_ Solve the following initial value problem: You can use Wolfram; Alpha to evaluate integrals, but must show solution steps.y +ry = ry y(0) = V2

(15 points_ Solve the following initial value problem: You can use Wolfram; Alpha to evaluate integrals, but must show solution steps. y +ry = ry y(0) = V2



Answers

Solve the initial-value problems in Exercise $15,$ and use a graphing utility to confirm that the integral curves for these solutions are consistent with the sketches you obtained from the slope field.

That would try and get it turns no return to determine the solution. Poor wide herbal crime minus why people toe wonder like he using. Why one e corn zero. And why prime off one You calling negative too. First thing is, first we have this part here with y double Prime Minister. While this equals year old general equation is gonna t call to see one you of he put seat you year minus he. This is for the monstrous proportionately patient that non homogeneous would be one of he e of t plus feet. Two of the e of negative t I think we do. We can write the formula here using o B one prime key. Yeah, he put feed you prime e of negative key ones. Euro be one time key he of he highs b a t e of negative t equals Equalling one divided by t. Then it will be the one off the he calling the prime of teas Equalling one part apply to teat e made the one key be want to immigration from line he each and negative you divided by you d you, I think into prime teas equal to Thank you. You hear? Keep one prime B two Time of keys equal to um Yeah, May s one t Yeah, yeah to Yeah, he to mass one teal immunization for one into the u E u. And you carry the whole wife quay on live TV equals two C one e t pussy. Two years minus t Plus you half times 1/2 times e f t integration from one of t e to the negative. You divided by you. Deal minus half e to the negative. T integration from one to t e off. Nick ut viper. You you Why one he crude to see, see? Want, huh? Hunt plus cto of e negative one, which is equal to zero si two is equal. T minus sea you you kill. Why? Prime of please equal to C one e a key c two e of mines key plus e a key integrations won t you have negative You divide by you d you plus one divided by Q she plus one q e of negative g integration from one of keep he a You differ by you You mais wonder I say e which is cool to see one of your teammates. See your brains, plus ah e the integration one e e mais by you, do you? Plus he of negative key immigration from one of e of you divided by you. And that will be simple. Why Prime one would then be si won t minus seat to e of my one And these canceled zeros because there once out of bottoms, that'll be equal to negative two. She won two times. He's equal to minus two C one miners wondered by e see to then become a here to the general equation. Wife key is he who, uh, e of t in creation from one of key e f minus you divided by u u minus half be of nice key integration you have you to u u minus the of tea humane for minus one plus you one minus t live to then become one negative e of one plus the names one. Would you be approximately negative? 1.9

Hello. Let's get into solving this problem. So what happened was that we plugged in question number 16 into a differential equation solver. We use the initial conditions and we got this equation. So let's try to plot it. Okay, though we confined is diverted. Also gonna put this Mormon X squared plus X to the minus two confined. Its diverted to be two X minus two acts to the minus three and we can put the directive equal to zero. So that would give us two. X is equal to two over X cube. These tools will cancel this ex woman transit X to the cube, so end up with exit. 1/4 is equal to one. That means X is going to be equal toe the four fruit of one which is going to be equal to plus or minus warm Ariel. So we know it's enough turning points and plus or minus one second. Well, let's look at the con cavity to help plot it. So we end up with a secondary to being to minus and just gives me plus six over exit of four, which which now we just need to check how it's con cavity is gonna be, but luckily we know for a fact that exit 1/4 will always be positive. So this is positive. So that means six a positive number or a positive number is always positive, that regard even better. So we're adding to which is a positive number to a positive number, which is really redundant to say that this thing is going to be positive. And because of that, that means that con cavity is always going to be upwards. Finally, we just know for a fact that this saying X cannot be equal to zero because of X is equal to zero. This equation breaks so we just end up with something that has a turning point at one in minus one, and then finally always is pointing up. So the best graph to indicate that would look like this. There we go and the area ass Antarctic Lee. It will go to infinity as exits. Costa cruises here because it cannot be closer, closer to zero. It will look like a normal car bull on every side, turning points against be at one and minus one

Hello. Let's get to solving this problem here. So we did was plug in the difference equation into a differential quick and silver and got this equation. Then what we did all screen is concrete. The first narrative and the second dirty, which is what we need to solve this problem. So let's take the first to return by the turning point to find a turning point. We just said why Prime equal to zero on that point is going to be when 1/2 need to the X minus 1/2. Is that mine hat Nader or its parts of negative? Yes. When I eat a minus, X equals zero, pull this over 1/2 e. T x is equal to 1/2 e to the minus X. We divide both sides by 1/2 which cancel these and then Allen Both sides and X is equal to minus X. And this is only true when X is equal to zero. So we already know The turning point is when X equals zero and I'm at that value. Why zero is equal to 1/2 plus 1/2 Zika one. Okay, so we know it's going to turn at 01 and it's only has one turning point. So let's look at the second derivative and check. It's come cavity, so input wide. Double crime is equal to 1/2 e to the X plus 1/2 eat the minus x. Yes, so here's a problem. This value is always positive. Either. The ex was always grated in zero. So meanwhile, for the want this part post 1/2 we can do is call this one over easy. So now if you divide a positive number by one, so one over a positive number is always going to be possible as well. So this part is also positive. So we're all we're doing is adding a positive number, plus a positive number, which gives us a positive number. So the con cavity is always positive, meaning it's always pointing up, meaning All we get is any curve that looks like this. We are one right here, zero right here on always pointing

But you are come to solve a problem. Number 19. So we have X y dash equal y los X word sign X eso We can rewrite that equation, boy. Uh, why does equal why does minus one over X Why equal X sign eggs. So we have the integration factor oil affects equals e to the power integration minus one over x dx. So it will be e to the power Negative length x So actually be equal e to support Len X miles to support my swan will be one over x so by Michael Blind Boy one over X gives that one over X Y dash minus one over X squirt. Why equal sine x So we can be right That one over x Why all that equal sign X thus one over x y equal Minus who? Zain X loss See planting condition So no Why equal minus x design X plus c x So get X since we have why Why equal zero? So by substitution Negative boy what a or a minus one Los See oy equal zero So c will be equal minus one. So why so why equal minus x Who? Zain X minus X Thanks for watching


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