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Solve the given initial value problemy" - 4y' + 3y=0;y(o) = - 1, y'(0)=The solution is ylt) =...

Question

Solve the given initial value problemy" - 4y' + 3y=0;y(o) = - 1, y'(0)=The solution is ylt) =

Solve the given initial value problem y" - 4y' + 3y=0; y(o) = - 1, y'(0)= The solution is ylt) =



Answers

Solve the initial-value problem.

$ 4y" + 4y' + 3y = 0 $,


$ y(0) = 0 $,


$ y'(0) = 1 $

Here we have to solve the initial value problem, which is given by nine y double prime last fourth white brain plus four way since he called to see you and the initial value, sir, Why zero is equal to one on white crime. Zero? Yes, he called to zero. So now we can associate the characteristic excavation. Tow this differential equation which kind of glutinous? 90 square plus 12 t plus four is equal to zero. So this is nothing but three t plus two. Whole square is equal to zero. So this imply it steams the call to negative to over three. Now we see that there is only one route in the captured stick excavation. So then the fundamental solutions to this differential commissions are going to be to depart negative tooth hurt X comma. It's times e to the poor. Negative to third ex to these are the two fundamental solutions. No, it's internal solutions. It's going to be Why is he called to see one times e to the bar. Negative to third ex plus C two times x times e to the power Negative 2 36 which can be also return us. You to the port. Negative tooth hurt X time, See? One plus C two weeks. So now our goal is to determine this C one and C two. So now there. See? We have Why zero is a call to one. This in place C one plus C two times zero Is he going to one? So the same place he want is going to one. We have found our Steve one. So let's take the derivative. Why Prime X? It's going to be e to the power negative to third X by the product rule. And then I have been negative. Tooth hurt time see one plus C two weeks plus you to deport Negative to third ex. I don't see too No, I have Why Prime zero is equal to zero. See if I take white prime zero, I end up getting negative to third time. See? One plus fee to is he called zero. So now I already found our C one is equal to one. So I have C two. Is he called to positive tooth hurt. So no lives right down the final solutions which can return us. Why is he going to eat to the bower. Negative two x over three time see one, which is one plus e to which is to over three times six. So that is a large bird. Equal a solution. Tow this initials a new problem.

Hello, everyone. This given question. We have to find a solution for why devilish minus y dash minus 12 eyes Jill. The conditions surviving is equal to zero and whitish one is one like so now First real light dogs related question for the visually question. So I'll square minus. I'll minus 12 is sequestered judo. So we get the root says one plus minus under one minus one plus 48. They were about to It just it was too one plus minus under 49 by two. So the roots off the equations is our request to fall and all that goes to minus tree, right? So since the WUSA riel and distinct So the general form general solution for the differential equation is see when you to develop four legs plus Saito eight by minus three X like Well, differentiate this so either. Taxes four c 1, 84 decks minus three. Saito double minus three X. Right now we have their player initial conditions. That is why one is Jeez. Oh, why one is equals to see 1 18 above four, plus c do it about minus tree. And why this one? If they questo one which is equals to foresee when a to the power four minus three C two A. To develop my industry. Right From these two conditions, we have to find a C one and C two. So when we we can from these two equations we get saved with request toe minus even $82. 7 Like from the first equation, we get set to us minus C when he took about seven this week and substitute in the 2nd 2nd the equation. So we get force even eight at about four. Plus, do you see one? A. To the power four is it was the one. So we get the value of Stephen as aided by minus four by seven and see to us it minus eight to the power three by seven. Like so, these are the values of C one and C two will substitute these values in the general form off the solutions we get by access. One day 78 double four X minus four minus one by 70 to cover Today minus three x Right, So this is the final solution for the initial value problem.

We have y prime plus two y equals one. Now, let's go ahead and figure out the integrating factor. Each of the two t now multiplying through by the integrating factor and then rewriting the left hand side using products role we get into the to T Y Andi to the to T. Okay. Okay. So again, we have multiplied through the differential equation by each of the two t and then we recognize the left hand side is equal to what I have written here. Okay? And now let's find the integral of each side. Yeah, divide through by each of the two t. That's it.

But we have y prime plus two y equals one. Let's go ahead and figure out the integrating factor. So now we're gonna go ahead and multiply the differential equation through by E to the to T. And then we're going to rewrite the left hand side using product rule. So the left hand side is gonna look like derivative of each of the two t y. On the right hand side is each of the two t. Let's go ahead and integrating side. This gives us this. Now we're almost done. But we still need to solve for the value of C. So this is what why is equal to We just need to figure out the value, see, and we know that why had 001 It's also equal to one half plus c. So we concluded that C is equal to one half. That's it


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