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Find the general solution of; J" + 2y' +Y = 8r?e'...

Question

Find the general solution of; J" + 2y' +Y = 8r?e'

Find the general solution of; J" + 2y' +Y = 8r?e'



Answers

Find the general solution of the following equations. $$\frac{d y}{d x}=-y+2$$

Come to this lesson and this lesson moves of or find a general solution of the first or the differential question that we have here. Now we'll ride this in a form. Yeah. Mhm. Okay. So we have P. F. X. That's one. Then we have care of X. Mhm. As two plus 2 weeks. So this would introduce an integrating factor. That is each of exports go to the integral P. Of X. Yes. So we have a chop X. Which is recalled to the anti growth one B. X. Which is recalled to X. Then now we have the Y. S. One over E. H. Of X. Than the integral. Uh huh. E. To the park H. Of X. Then times Q. Of X. Dx. Okay. So at this point why is he called to one over E. To the power X. Then the integral. Eat to the bar X. Times two. The last two x. E. X. Okay. So at this point we have mhm. We'll bring a negative. Yeah. Then we were split into two. So we have one that is two E. S. And we have another that is okay. Two X. Eat the bar X. B. S. Okay, so let's solve this bad. Then we have the solution of the path to it. Mm hmm. So we have two x. Eat the bar X. Dx. Yeah. Yeah. Yeah. Oh yeah. Now less equipped this too. The integral S. The teeth. And this would be S T minus the integral tds. Using the suppression. Uh Using the the integration by parts. Okay. So with this we would yeah, it quits S. Two two X. Then would equate the D. T. To eat the bikes. So this implies that uh D. S. Is equal to. Oh quick. This implies that idea says record too two. Okay. Mhm. The X. Than at T. She got to the so T. As he called to eat the bikes. Because when you take the to grow both sides you still have heat the bikes. Then the whole of the integral to X. Each of the bar X. Dx come Britain us as times T. So access to X. Then T. Is E. To the par. So this becomes two X. E. To the X minus the integral T. And now dx is to the X. So this becomes too. Yeah. Okay so this is what we have for the whole of this so we can put that in there. Why is he called to E? We have this Square root of two x. DX. Then in place of the whole of this who put will have had here just a positive two X. E. X. Then my nest into ground to e meetings. So we see this as the same thing as that so they can cross themselves out. You have wider difficult to eat. Oh And this is Times two x. Yeah Bladders see Yes Plaza Constance Always one. We integrate and uh the balls are not fixed. We bring a constant now that we know the to this and that will cross themselves out. Yeah. Why that is she called to two X. Last see E to the negative back. So this is the general solution for the equation which was given to us. Okay. Thanks for your time. This is the end of the lesson. Yeah.

So here we are given the second order differential equation. Why prime prime Plus two of Y Is equal to zero. And writing the characteristic equation here we have arab squared Plus 12. A constant is equal to zero. So remember here that We don't put an hour when there is no derivative of why? So here since we just have a Y. It's just 12, not 12 are now going forward serving for the roots of this equation. With the characteristic equation. We have r squared, our squared being equal to minus growth. And taking to the rules both sides, we observed that we are going to have complex roots and they will be plus or minus For the root of -12 which is plus or minus. There were 12 4 x three x -1 which is minus proof. In here. We have plus or -2 Who drove three i. and these our roots. And now our general solution will be in the form of why T. Is equal to C. One E. To the power of lambda T. Of course new T plus C. Two E. To the power of lambda T. Sign, new T. And we're lambda zero here. And our mule is 203. Now substituting this into our general solution. From here we have whitey Is equal to C1 and since we have E. To the power zero T. The E to the Power zero becomes 1. So we don't have an e term here. So of course our new is to root of three T. Right? Plus C. Two. Sign To write off three T, and this is our general solution and our final answer.

First, my pride. Why, Yeah, If Why You waited you. Artie, you are Here's one.

Hello welcome to this lesson. In this lesson you will find the general solution for the first quarter differentiate question why prime minus two Y. Is called to one. So compared to a general form why prime glass P. Of X. Y. That is he called to care of X. Oh and here if you are comparing the P of X would become negative tube And care of ex would become one. Okay. All right okay. At this point we would introduce an integrating factor each of us that is recalled to the integral of P. F. X. Dx. So this is he called to Integral of -2 DX. Which is called to -2 X. Then we can write why us one over E. To the part H. Of X. Times the integral E. To the far east of X. Than time scale of X. The X. So we have weather is equal to one over. Eat the bar negative two X integral negative. Uh Eat the bar negative two X. Times one the X. Okay so this is the call to eat the bar two X. And if we differentiate if we integrate this we have negative for an on to E. To the negative two X. M. Plus the constant C. So why is he called true negative half At this time that will become one. So we have plus See then eat the part two x. Okay so this is the general solution of the differential equation times for a time this is the end of the lesson


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