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(8 pts) 13. Find general solution to the differential equation y" +Ty" + 16y' + 10y = 0(11 pts) 14. Find particular solution to the differential equa...

Question

(8 pts) 13. Find general solution to the differential equation y" +Ty" + 16y' + 10y = 0(11 pts) 14. Find particular solution to the differential equation+5 G +6y = xe' .

(8 pts) 13. Find general solution to the differential equation y" +Ty" + 16y' + 10y = 0 (11 pts) 14. Find particular solution to the differential equation +5 G +6y = xe' .



Answers

Find the general solution to the linear differential equation.
$$
8 y^{\prime \prime}+14 y^{\prime}-15 y=0
$$

Okay, we're getting three. Why? Double time? Minus 14. Why? Crime? Bus mates. Why is he still? But you write that as greedy. Climb this grade when it's 14 and the state you could still using. Our quadratic formula gets 14 plus minus is graybeards. About 14 squared. What is for a C all over two times A. This is equal to 14 plus minus 10 over six till he gets to pollution's. That's four and two over three. So we have our characters equation. Why people to see one e t o four ex first see to each other to over three x.

We're going to solve a problem. Number five. So by comparing the governor collision wife dash waas on equal one with general food. Why does Why best less you've x Why equal you x so way See that you fix equal one. And the integrating factor is I Thanks equal. Yeah, so far integrate be off X doing x so it will be equal you Which one? The X. So what? E to the power X So by multiplying the differential equation by FX So it will be, um e x loss you x equals e x So the French prison will be Why dash X multiply Why dash e x multiply? Why so e x multiply one So it will be Yeah. Okay. Why Bess April 2 so far x so x y equal integration. He supports x the X So it will be Well, um e x y equal. It's about X plus c. So will be equal e sport x y if support x by support x he just four x over here for X waas, see over divided by details of our techs. So the final solution will be equal. Why equal one lost C e news? Um, the power minus x Thanks for watching

Okay, So in this problem here, we need to solve or find the general solution by first finding the auxiliary creation p of our and this is going to be a two so r squared minus 14 R plus 58 like So now we set that equal to zero. So let's try to ah, find Ah, Let's try to factor this first. So the factors of 58 here eso if we do 1 58 that does not add up to 14. Now, let's try to And, um so 58 is to end 29. All swell. Um, so that does not factor into anything else. So to 29 is 31 s o. That means we'll just try to use quadratic equation to solve this, right? So that's going to be equal to our, uh, is the platoon negative B, which is 14. All right, 14 plus or minus. So B squared. Ah, 14 squared is going to equal to 100 and 60 aids. And let me 14 squared first. Uh oh. Sorry. 196. It will be 196 minus. And then four times 58th. Let me put that in cash four times a T. It is 232 here, remember? That's Ah B squared minus four a. C here. So 1 96 minus 2 32 It's 2 32 gives us negative 36 here. So that's going to be Ah, square root of negative 36 is going to be six. I, ah, six I and then we have divided by two is three. So are here is going to be Ah, 14. Divided by two is seven plus or minus number This was six. I divide by two So plus or minus three, I hear. So then, uh, now that we have the these to ours, then our general solution why of X? It's going to be able to see one e to the and then we'll take the real part. So that's seven x seven x and then sign of three x So that's the This is the Amash in a part, plus C two e to the seven x co sign three x. So this is our general solution here

All right to solve this differential equation. Let's first write it in linear operator form. So that's gonna be the square and minus 14 D. That's in privacies plus 58 of why is equal to zero. So then the corresponding differential or sorry, the auxiliary equation is on the Beagle two r squared minus 14 R plus 58 like so? So we said that he was zero. Now, if you want to try to factor this, we need to take a look at the factors of 58 that add up that add up to 14. Okay, so the factors of 58 include ah one and 58 which clearly does not add up to 14. Next is going to be two plus 29. This also cannot factor into anything over. This does not add up to 14 and cannot factor anymore. So we're going to need to use quadratic formula here to solve this. So we get our is equal to, and then we have negative B, which is going to be plus 14 plus or minus plus or minus and then square root of 14 squared minus four times 58 all over on then to so out front here. First we have seven. So 14 divide by two is gonna be seven, and then it's gonna be plus or minus this. Um, 14 square is 196. All right, so that's 196 and then four times 58 is 232 somewhere in a 232. So then now this becomes negative. 36 little things to square to that is going to be equal to six I. And then we have divided by two on the bottom. So that's gonna be plus or minus three I on the bottom. So this is a here. This is be here. So our general solution Why of X is going to be able to see one e to the A. X. So seven X, then sign of B X. So and then plus C two, e two, the A X again and then co sign of B X, which is cosigner three X here. So this is going to be our general solution here


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