Solve the initial value problem dy0) = ~i)yo)With Y(O) =...

Solve the initial-value problem. $$y^{\prime}+y=e^{2 t}, y(0)=-1$$

Hello, everyone. Today we are going to sort problem Number 34 this is we have to find it in this leper stand for that is you have an initial value problem. The given our by double s my last life because you were not be. And while I was little it goes to And why would I sell zero? It goes little. So taking leftist transall or boots is as a square by all of us. Quietness says, while I was little my less by myself syrup my enough well I office because that is you learned after that is fair play. It's OK, shift in India. Very good. You didn't bar minuses, but yes, Why are zero is too. Where myself zero is cereal so combating We can try to test my office equals he Mr Minuses. But as in the as a wrestler in the s minus 11 Bless minus two s or my coastal right and second chance place Onder there be as Pressler in the s my uniform. So by playing partner fashion with 30 for this and it's for you can write business and by us for us B by us. Let's learn plus c by as Max on this kind Return us. Do you buy? Yes. Bressler left you by F minus. Learn so it can be forward Breath blinded by as a square minus one. Some stewed as inconsiderable. That is usually we get the nest. My enough like big us one by his Bressler that you can be found ordered by one by yes, Indo as my substitute as incomes minor form. It's a big getting us like by foot. Similarly See is a good one by s Indo. If I flirt Substitute Asik Whistler were bigger tinnitus one by similarly the because us there by that's minus lock Some stood as because my but because to less developed by yes pleasure. Some stood a single one which is in Cordoba. Work, Phil, my office and return us my enough one. But as you know, the border by an SS less one by you Did a par minus sense there, right by as this long plus one by do you does a par minus is they aren't but s minus one Bless learn by s press on press learn by AZ minus up. Do you think you can be sure? Followed by applying so good shifting Jenna if because of Futebol bonuses they were very impressed by the metaphor. And my dear brand by you're enough big So our biases the you are not take plus one bite contributors to minus T Do you think best buy do you, my Nesler Inter, You are enough tea. I'm not late plus one by two certainly in the u necessity. But he added Will be day My Nesler, You are Loveday Freeman and things is minister minus T plus the list that if by often that wasn't awful question Thank you.

Okay. So for this problem let's go ahead and start off by replacing the wise with ours and said. So why does the prime translates to R squared? Why prime transforms our And why it transforms into one and here I don't see any obvious factoring. So the factor let's go ahead and use the quadratic formula. So I have negative B plus or minus the square root of B squared -4 times a times c. All divided by two times a. Get some fi down we'll have negative one plus or minus the square root of negative seven All divided by two and we have negative one half plus or minus I route seven divided by two. And so from this we can actually build our solution. So your solution is going to be the form C1 Co sign of Route seven divided by two plus C. To sign I've reached seven divided by two all times E. To the 9 to 1 half. And we can plug in our initial value conditions to solve for. So you want to see too. So our initial valid conditions are that C. One R. Sorry that wife of zero and Y. Prime zero both equal to zero. Let's go ahead and plug in wives zero. So we have zero equals two. The exterior right here. Sorry there's a next term. This exterior is going to go to one and then this right here is gonna go to. Sorry, I forgot the X terms. So when you plug in zero right here sign of zero is 0. And then on the left hand side we have co sign of zero. Co sign of zero is one. Yeah. So we have c. two. We're sorry C one plus zero. So we have c. one equals to zero. And that allows us to simplify down the equation to B. C. To Sign of 2 7 divided by two x. Times eats the negative one half X. And now we can go ahead and take the derivative. And when we do we get wiped prime equals to what we have a product rule here so we'll have negative one half, eats the negative one half X. Uh Sign of Route seven divided by two x. And there's a C. Two here. So good here plus C. Two. Uh huh. Co sign Of Route seven divided by two x. Times. Route seven divided by two Times. Eat the later 1/2 x. And so when you plug in our initial value conditions with this case which was Why prime of zero equals 0. The left hand side and the right hand side. So for the left hand side we're going to have um zero And on the right hand side we have c. two. If you plug in zero here we're going to have zero. So C2 Route seven divided by two. Uh It's a negative one half, which is going to be one, And in this case C2 is also going to be zero. So if C one and C two are both zero, our solution for this differential equation is actually going to be Y equals to zero.

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