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Find the solution of the given initial value problem:ty + 3y = t2 _ t+5, 9(1) = 7, t > 0...

Question

Find the solution of the given initial value problem:ty + 3y = t2 _ t+5, 9(1) = 7, t > 0

Find the solution of the given initial value problem: ty + 3y = t2 _ t+5, 9(1) = 7, t > 0



Answers

Solve the initial value problems. $$\frac{d y}{d t}+2 y=3, \quad y(0)=1$$

16 to number 15 were dealing with linear first order ordinary differential equations in this section. So pretty former formulaic. If you can get that we need standard form. So do I. D t plus a function of x times y equals another function of X And in the key to solving this would be to multiply both sides by an integrating factor as such. So in our case, this p of X is going to be too so e to the integral of to, um, in the space it's gonna be DT um, enemy just or consistency. I mixed up my exes and my tease here again, this is a function of t function of T when you integrate to with her sector. See, that's just going to be e to the to t. That is my integrating factor. So if I can multiply this entire equation by E to the to T so I would hav e to the to t times d y t t plus two e to the to Tito and why is equal to three need to the to t. And that allows this left side to be written as the derivative of a product That's the derivative of E to the to tee times. Why? So the derivative of E to the to tee times why is equal to three e to the to t Now the key will be to integrate both sides of this equation. So you get why e to the to t is equal to when you integrate this. You have e to the to t and then divided by two plus a constant of integration. And then my initial value problem was that wives zero is equal to one. So in solving this for why, why is equal to divide everything by E to the to T You're going to get what three has, plus C e to the minus two t And then when you, um uh, substitute, why of zero is equal to one that would give me one is equal to three halves plus c so 1/2 minus three have remind it's 1/2 So C is equal to minus 1/2 in this case. So my final answer is why is equal to three halves and then minus 1/2 e to the minus to t. And that is the solution to this differential question.

Falling in differential region with the initial condition that Y zero is equal to dignify. So this initial condition is gonna allow us to determine what the value of C is after we take the indefinite in trouble. So the person that you do when you have a depression region that separable is you get the different variables on opposite sides. So are two variables here tea. And so we get them separated. It's to multiply by t. Then would you have me? I've is equal to three minus to t 80. Um And so now we're ready to take the integral of both sides. So get the injured girl. You Why this sequel to into role of three brightest to t beauty. Although this is just equal to y uh, so why he's equal to three t minus t squared and we're just doing the reverse Powerful. You have teach of the first hours that becomes t squared over two. But then the choose cancel outside Justices T squared of enemy have to remember to add plus c So we've essentially solve this right here this differential equation. But we're not gonna plug in this initial condition or to figure out value of seat. So we know that when, Why, when you plug in zero, you get negative. Five. So negative five is there? Why? Value and not equal to three times zero minus zero? Let's see, That means that negative five these people to see It's an asset. We know the value of seeing. You can just plug that into our solution. So why is equal to negative T squared plus three tea minus five? And this is our solution for the initial conditions that were given.

The only statement is the way upon d d. He waas nine years power 12 minus three D And given that why you're full equals seven indeed early. Why he was indeed a nine unit power drill minus three D duty. So why do you have the equals? Nine. Few devour. Do you have a minus through D upon my bestie class c, huh? No. Why You have d his equal. Do my ministry. You, Bauer, Do you have a minus three D plus C? And why you 47? This is equal to my honesty. You baba. You know, Let's see. This is seven. There's equals minus tree plus c Therefore C equals 10 hands. We gate. Why you have d he goes minus t You power to have minus three day plus 10 as the degenerate era, 10 city system. And so

16 12. Problem number 17 This is a first order linear, ordinary, different journal equation. We have a standard form, but before I do that, I mean, you might get your You've done enough of these. You recognize what is the derivative with respect to theta? Why times data? If I'm differentiating with their sector fate, I would have to use the product rule. This would be you take Why them's the derivative fatal with respect to fade as one plus the derivative. Or why, with respect to theta time Stada. So this is what theta de y de Theda Plus why you notice that that is the incomplete left side. So I could take this and say OK, well, I can go straight to the derivative with respect to the data of y theta is equal to sign data. I can go straight there and start solving this problem. Okay, if you are getting toward that is not jumping out at you. You're not recognizing that we have a very methodical formula for the way we've solved all the differential equations in this section. That is to get the problem in standard form, find the integrating factor and then proceed. So let's just say that we did not recognize that shortcut. Standard form in this case is going to be I need to divide everything by theta so d Y d theta plus one over theta times. Why is equal to sign of Fada over theta? The integrative factor is going to be e to the anti derivative one of earth. Ada di Data This is e to the natural log Absolute value Fada. Since that is positive, I can remove the absolute value signed. This is E to the natural log of theater which is theta so everything has to be multiplied by theta. You would have the derivative with respect to theta of Fada. Why equal multiple Beth data and you get sign of failure? So this is where we were a moment ago. Now integrate both sides of this equation and you end up with why Time stada is equal to minus coastline data plus a constant of integration. So we've got wife ADA is equal to minus co sign data plus a constant of integration. The initial value problem was that why at pi over two is equal to one. So if why a pi over two equals one that tells me that in this equation So first of all, let's just solve it for theta. Why is equal to minus co sign data or with eight A plus CIA Rotana? And then if y harbor to is one that tells me one is equal to minus co sign higher or two over pi over two see a repair were too co side of PIRA to is zero. And so this tells me that one is equal to to see over pie. Therefore, C is equal to pi over two. So this final answer is why equal minus co sign data over data plus pi over two data.


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