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Me Mo loBIISoruB OSolve the initial value problem XI= AXIXQ) here UBTi3 sin Stli4icos 5t WHMA Xce) em L"3 cos St _ + sin 54][3 sin 5t 4 cos St| B.XE) em U3 cos...

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Me Mo loBIISoruB OSolve the initial value problem XI= AXIXQ) here UBTi3 sin Stli4icos 5t WHMA Xce) em L"3 cos St _ + sin 54][3 sin 5t 4 cos St| B.XE) em U3 cos St 4isin 5t[-3 sin 5t cos ISe] Xit)l ex 13 cos St 4 sin 5tsin 51 4 cos St] cos S0 4 sin StD,Xl)Seclml SifirlaBolum 4 / 10 0-

me Mo loBI ISoruB O Solve the initial value problem XI= AXIXQ) here UB Ti3 sin Stli4icos 5t WHMA Xce) em L"3 cos St _ + sin 54] [3 sin 5t 4 cos St| B.XE) em U3 cos St 4isin 5t [-3 sin 5t cos ISe] Xit)l ex 13 cos St 4 sin 5t sin 51 4 cos St] cos S0 4 sin St D,Xl) Seclml Sifirla Bolum 4 / 10 0-



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Solving initial value problems Solve the following initial value problems. $$y^{\prime}(t)=\sin t+\cos 2 t, y(0)=4$$

They want us to find the solution to the initial value problem And we can take the indefinite integral on both sides So we'll end up with negative coast. I don t What sign do you think? Over to for someone on constancy One What? We know that at zero we should get four So we'll have negative one. But Syria we'll see when is equal support where that C one is equal to find so that I have t is negative co signer team plus 1/2 time sign of duty Ah!

In this problem, we're given a fort ordered narrative off at lunch. They were given points where the birthday of its best mystery 2nd 1st and function itself as through and where has to find the function itself? One. So let's do this. Let's first take tea in China off the court ordered evident. Fine, deeper ordered aired. So why burn order will let me negative science t co sign T e t. Well, we have You're saying there said And this will be, uh I'm sorry. That will be then co sign t plus sign t plus some constant C one and we will use the condition for a negative on the Constant. You know that third narrative is seven. What a tease. Zero. So I got this coastline here Go plus sign zero plus c one. You know that the sign zero is zero and coastal zeros want soda. C one should be six now. From this we find hurt David too to be co sign T plus sign T plus six less fall the same procedure. Let's take the triangle office to find the second order there. Why do you Then will be integral comes Santy plus sign T plus six d t. And then this one will be signed. T minus tose I Inti plus 60 plus C two. Now let's use once, given we know that the secondary table me negative one won t zero So that is Sign zero minus co sign zero plus zero plus c two saying From this we know that the size here 00 is one. So we have will be over in hand negative one here so that I see two should be zero and then we can write the second order to be, um, sign t minus Go side T plus six team, Same procedure. Let's take the struggle of this one to find ah y part of the first order, dammit! That only scientist e by the scope sci nti plus six tvt And that will be negative. Of course, I t negative or mine Society plus 60 spread over two for some persons. Three. Again, let's use the given point. We know that the first words will be negative one when up he's zero so that it's negative coats. I'm here when inside zero plus zero plus C three and sign 00 Soco says he was once again we have already won so that C three should also be zero. So then the first director, why primal dysfunction is negative co sign t my science t plus three t script. Now let's take the internal bits to find function. We know that. Why? Well, then the integral off negative goes and t minus sign t plus the spread E t uh, this and will be negative. Offsay Inti Plus those scientist He was three t cute over three, plus a constant c four. Let's use the given condition. We know that why will be zero in peace zero So that it zero negative side zero plus co sign zero plus of zero plus C four This is zero co says your eyes one in order to zero left inside c four should be negative on And it didn't look that we didn't find the function to be. They get their hopes I Inti plus go society waas trees Guess love t Q uh minus four. So again we have negative society Moscow Society plus T cubed minus work

We want to solve this initial value problem here. So this is just a fancy way of saying we want to find some function s when we take the derivative of this with respect to T, we will get one plus co sign t. And when we plug in zero and two or visual function, we get output of four. So this over here just saying we want the 0.4 on it. So to get us to appear, we can go ahead and integrate each side of this with respect to t. And so that's gonna give us s on the left hand side and on the right hand side. Well, we can apply the linearity of our internal to rewrite this as integral of one plus the integral of co sign T d t now integrating a constant. We just multiply by what we're integrating with respect to which is t so this is going to teeth. And then if you look at that anti derivative chart that they provide us, the anti derivative of co sign is just going to be sign. There's gonna be plus sign tea and then we have our constant C so This is our general solution. But we don't care about the general we care about. What we get will be plug in zero and we wanted out. Put a force. Let's go ahead and plug that in salt for C. We have four is eager to so zero plus sign of zero plus c will sign of zero is also zero. So we just get or is he going to see? Don't go ahead and take this. Plug it into our general solution and that tells us that s is equal to t plus sign of teeth plus four or our solution of this initial value problem.

We want to solve this initial value problem here. So what we're gonna do is integrate each side of this first. And by doing that, we'll get S is equal to the integral of co sign T plus the integral of scientist, and that's using the linearity of it. And the reason why we're doing this here is because we're looking for what function s derivative is co sign teeth society. And it outputs one. When we plug in pies, that is really just saying pi one is a point. I was going to interpret each of these. Now, if you look at the anti derivative chart that they provide in this chapter, they say to integrate co sign, that should just be sign, but then to integrate sign, that should be negative co signs. We have negative co society and then we add our constancy. So this is our general solution. But we don't care about the general. We care about win. We plug in pie. We wanted output of one. So let's go ahead and plug this into here. We have one is equal to so sign a pi minus co sign of pi plus C now sign up. I zero co sign of pies, Negative one. And now these negatives will cancel each other out. And so we're going to have that one is equal to one plus c. So if we subtract one over, we get zero. Is he going to see? So we go ahead and plug this end to hear now, and that's going to give us that our general solution should be s is equal to sign of T minus co sign of tea.


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