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Let y(x) be the solution to the initial value problemy'' + 4y' + 3y = 20e −2x sin(3x), y(0) = 5,y'(0) = 7.Find the value of y(1). (In Radi...

Question

Let y(x) be the solution to the initial value problemy'' + 4y' + 3y = 20e −2x sin(3x), y(0) = 5,y'(0) = 7.Find the value of y(1). (In Radians)

Let y(x) be the solution to the initial value problem y'' + 4y' + 3y = 20e −2x sin(3x), y(0) = 5, y'(0) = 7. Find the value of y(1). (In Radians)



Answers

Find the solution of the initial value problem. $\frac{d y}{d x}=\sin x, \quad y(0)=3$

Hello. And today we're going to solve the initial value problem of why, To the fourth derivative equals negative sine the T plus co sign of tea. And to help us all this were given three conditions. We know that why triple prime of zero equals seven. We know that. Why double prime of zero equals wide prime of zero, which is negative One we know that. Why zero equals zero. Okay, No, First of all, to soul of initial values problems, you always want to think about integration. So we want to try to integrate all the way down until they get to get can get to a wide of zero because we want why undifferentiated So gonna take the integral off. Why the fourth derivative equals the integral of negative sign t waas co sign Jeanne you We always want to have a DT in front because that's our integration factor. Basically. So now you wander riddle way. So why should third derivative equals co sign of tea cause the derivative of negative but sorry. The integral of negative scientist is ko 70 and then plus sign t plus C, which is our integration constant. So once we get this. We want to plug in zero for why? Triple prime? No, my bed. We want to pull you in 77 equals co sign of zero plus a sign of zero plus c So seven equals one plus c. Which leaves us with sea eagles six. So once we do that, we can plug in six. Foresee. So we have why Triple prime equals co sign of key plus the sine of t plus six. And I want to integrate again to get second derivative A Why? So we have why second derivative equals sci fi minus co sign of tea plus six t plus c our integration constant calls from our DT. So now we're given the condition that why double prime a zero equals y grams euro equals negative one. We're gonna plug in the negative one equals sign of zero minus co sign of zero plus 60 plus to see. So these cancel so negative one equals negative one. Percy, So C is zero. So since we know that CIA zero, we do not have to add any more turns to our Why double Brian. So why Double prime is going to give us sign of tea minus co sign of tea plus six t. So we're gonna integrate this again. Do you find do you RDX at our integration factor? Why prime equals negative co sign of tea minus the sine of t plus three t squared plus c. So now, since we're given that why prime of zero also is equal tonight a one can also plug that in so we can have negative one is equal to negative co sign of zero minus sign of zero plus three times zero squared plus c. You know, this is zero and we know that this is Europe. So we get negative one negative coastline and year old negative one negative one plus c c is again well, zero. So again, we do not have to add anything were the last time we're going to take the integral of white crime which is people to the integral of negative co sign but t minus sine of t plus three g square integration constant. So we are getting why equals negative sign of tea plus the co sign of t plus tty cute plus c so we know that why is +00 So when Weichel zero t is also zero. So zero equals negative sine of zero plus co sign zero plus zero Cute close to see. So this is zero. So we know that zero equals one plus c. So C is going to be negative one. And so we're gonna plug that into her final answer, which is going to be Why? Sorry about that. It's gonna be Why equals negative sign of tea. Plus the co sign of tea, plus t que minus one. And there you have it. Here is the answer.

So here we want Teoh. Um Sau this initial value problems were given the differential equation. Why? Prime is equal to why Cube sign of X okay and were given the initial condition were given that why? Of zero It is equal to zero. Okay, so well, we have Why prime? So why Prime is the derivative of wise. That's d y dx. Okay, And that is equal Chew. Well, why cubed? Um Why? Cube Sign of X. This is a Y cube. Sign up axis writing. Um why prime as do I d x. Okay, so then we have while separating our variables, we have that d y over y cube, right? Just implies that d why? Over why Cube is equal to while sign of acts. DX is playing both sides by basic DX and then the vitamin D i y que de y over y cube is equal to while sign of acts. Um, dx. Okay, then we can when we can integrate both sides. So we have the integral of while d y over y cubes. That's the integral of one over y cube for the integral of one. Over. Why Cube, do you? Why is equal to the integral of sine of X dx. Okay, so, um, so what we get while the integral of one overwork you just the integral of wide to the, um this should be Yeah, this is Yeah. This'd the integral of one over. You know, of one of White Cube is the integral of why? To the negative three power. Right. So, um, what we get here, we integrate, We get Why? To the negative. Well, to the negative three plus one. That's a negative thing, too. But I don't think this one, um, over while over negative three plus one. Right? Um and that is equal to the integral of sine of backs. Weston. That's negative. Oh, sign of acts. Plus a constant. Okay, so then using, um, let's see, um, so, yes, I mean this. I mean, again, you put us as why was why toothy there to Right, But this is justice issues in the formula, right? The integral of X. To the end, the X is equal to or x to the n plus one and then divide through by n plus one. Right. Okay. So, um, again, if you're doing these, you know, different equipment problem if I know how to do basic integration, but it's it's always good to kind of refresh. Okay, So, um, so what we have now is well, we have negative one over choo. Why? Squared is equal to negative co sign of axe, and then don't forget the plus the constant. Okay, so we can simplify this and get well, um, I mean, both negatives weaken divided by a negative one. And get well, one over Chu y squared is equal to while co sign a max. Okay. Plus your constant Okay. And then well, we that we use, we use our initial condition. Remember, our missile condition is that why zero is equal to zero? Please plug in. Well, um, when Why is when? Why is zero we have 1/2 times zero squared is equal to co sign of zeros. We have one over while, too. A c time. Zero square, which is two times zero, is equal to co sign of zero. Um, plus a constant right. That implies that Well, Corson is there is one. So we have This implies that one, right? This applies that one plus c, um, or is equal basically is equal to infinity. Right? Um because 1/2 times zero is basically and this approaches infinity. So if one plus c is basically equal or approaches infinity, then that implies that See, right, that that implies that implies that c must be must approach and finish off my C must, in a sense, equal infinity. Okay, so substituting c equals infinity in well in one over. Chu y squared right equals ah co sign of eggs plus infinity. Okay, So when c equals infinity, the solution becomes what it becomes Why, right? So when c when si is equal to infinity, we're gonna solution becomes why of acts is equal to zero. So we go. Thank you.

In order to determine the interpreting factor, we must first put it into the standard form of the equation. Ever Words must divide both sides by the coefficient of white prime, which in this context is X because we are exported prime. So again divide all the terms by this value to give us this. Now we know our p of X is negative one over X. Therefore, integrating factor each the negative times integral of one over X D acts gives us eat the negative natural of axe E to the natural log is one which gives us negative one times acts which in this context would be X to the negative one, which, in other words, is the same thing as one over X to the power of one which is just one over X. Therefore, multiply both sides of the differential equation by one over acts. We end up with one over acts. Why is equivalent to the integral of sine of axe de luxe? We end up with writing this in terms of why we know the integral of sine of X is negative co sign X and remember, we want why by Self Smiths divided to the terms by one over X we get Why is negative x co sign acts plus C times acts now are factor off Why have pie is Europe In order to solve for C, we plug in what we have zero is negative pi times co sign of pi plus C times pot As you can see, I'm literally playing this right into the equation we just found we gotta solve for C We underfoot see his native won Our final equation solution is why is negative acts co sign acts minus X.

This question have to find the solution of the differential equation and the differential equation is given to us. That is It was too B Y Y B S. It was true cynics delighted by of course. Why? Okay. And it is also being told that you have come find that solution for Why is 0? It was too late. First of all, I am going to separate here variables. Means I'm going to cross multiple life. If you cross multiply, then we get here. Caused by the way it was too cynics be. It's okay. Now I am going to integrate both the sides. If I make your integration means integration of course. While with respect to levi and integration of cynics with respectful VX. Okay, now we know that integration of course, why is signed white And integration of cynics is negative. That is minus corsets. And we have to had a constraint because while we are doing indefinite integration then we have We're always a constant. Now we can say that this joint glutinous sign away plus course X equals to see. Okay. But this seems a general solution. And now we have to find the specific solution here. It is being told that we have to calculate the solution for Why zero it was for zero. Which means that you have to take X equals to zero and why it was 20 in this Solution here. Okay. So I am going to put these values in the general solution. So if I put these values here, you can see that you get signed by main sign zero plus core sex means call zero equals to see. Now value of science zero is zero. Value of +40 is one. So I get value of say that is one. Okay. And I put or substitute the value of the sea than I get the specific solution that is signed by Plus four sex equals to one. Okay, this is the solution of this differential equation. Sign Y. Let's cause X equals to one. Okay, Thank you


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