5

4. Which of the equations given below is the solution of the following differential equation? (13 Puan) p? _ 9p + 18 = 0 (p = 3)Ov =3C1 - [ (y - 6x + C)l - 3r+C) = ...

Question

4. Which of the equations given below is the solution of the following differential equation? (13 Puan) p? _ 9p + 18 = 0 (p = 3)Ov =3C1 - [ (y - 6x + C)l - 3r+C) = 0Oy = 6Cx + [ Ov = 3Cx + C26 - 6r) ( - 3r + C) = 0

4. Which of the equations given below is the solution of the following differential equation? (13 Puan) p? _ 9p + 18 = 0 (p = 3) Ov =3C1 - [ (y - 6x + C)l - 3r+C) = 0 Oy = 6Cx + [ Ov = 3Cx + C2 6 - 6r) ( - 3r + C) = 0



Answers

Find the general solution for each differential equation. Problem 1-12.
$$3 \frac{d y}{d x}+6 x y+x=0$$

So we wanna find the general form of the solution to the differential equation. Do I D. X plus three y equals 18 in our method for solving these especially wanna check that it's in this form, uh, t y t x plus p of x y equals Q Becks. And when you can write the equation in this form, we're gonna use, uh, this integrating factor or we just take e to the integral of p of x, t X. And so, In our case, this is just e to the integral of three D X, which is just eat of the three X and then to solve equations of this form. What we do is we take this integrating factor and starting with this equation, the differential equation we have, you multiply both sides but the integrating factor, and so will get either three x d y t X plus three. Each of the three x y equals 18 either three X and now the whole reason of doing this is on the left. What we actually have is the derivative which will write Capital D, said acts of you to the three x or integrating factor times why you can check that This is the case by product roll. And now, um, what this means is that we can integrate both sides and on the left will get just what was inside here. He did three x times. Why answer is gonna be the integral of our expression over here, which you might want to use. U substitution to check, I believe should be eat of the three x plus c right when we take the derivative, we just multiplied down by three when we get 18. Even the three AKs. Okay, so we have this expression. It's not quite what we want, because on the left, we have, you know, three x times. Why? So to get why we need to divide by either three AKs. And so this term over here, we'll just be six that, you know, the three actual cancel out and then we'll get see, divided by Eagle three X, which is the same as e to the minus three X. And so this is the general form for our solution.

Solve this initial value problem, we're gonna use an integrating factor. And the equation is already in the right form. So to get our integrating factor you remember we have to do is just set this equal t e to the integral of the function in front of why on the left consort, integrating factor is just gonna be eat of three X and so to find why we're just gonna multiply our differential equation by each of the three X, we will supply the V x times eat of the three X. Remember, we have the same base, the exponents ad. So this will be 16 each of the four X. Okay. And then remember, our trick is that on the left hand side, this expression is actually equivalent to the derivative uh, you the three x times. Why buy the product rule? So then, if we integrate, both sides will get eat of the three x times. Why equals the integral of 16 eat of the four X with respect to X, and this should just be for each of the four X. Well, see, um, you might want to do you sub, but it's pretty soon to check that this is right. When we take the derivative, we just multiplied by a four. So then, to get why we'll just divide everything by e to the three x. This will get four of the ex plus c e to the minus three x. Okay, it's the last thing we need to. Dio is used the initial value to find C. So if we plug in X equals zero, we should get 13. So 13 is four times E to the zero plus c times E to the zero you 20 is one. So just get that 13 is four plus c, so C is equal to nine. And substituting that value in for C will have the solution to our differences equation.

Now in this case we're going to take a look at uh let's see here looking at a bunch of differential equations and I'm going to do all these next few problems together because they're all very similar. So we're given some difference or equations and I suppose I should have written them down for each problem. So let me actually, so 36 we have S double prime plus four S crime. Let's see. S +40 Um Yeah and again this is notation is getting bad because again engineering this would be C. And this would be K. Um So but they B. And C here for generic differential equation. So we'll keep with their notation. So basically they ask for what find their values to see that make the general solution over damp. Under damped and critical. Damn well the what you need to look at is the discriminate except to discriminate anyway. Yeah, I think so. The thing under the square root in the quadratic formula. So b squared minus four. See these all have a in you know in the quadrant in the characteristic equation of one. So A. Here is just one so B squared minus four C. In this case we have B. Is four, uhh C. Is C. And so we need we have 16 minus four C. Is that greater than equal to or less than zero? So if it's if it's um over damped it's going to be greater than zero. Well let's start over damped. It is going to be a greater than zero. Why did I? Oh I think I looked at them. These are probably all messed up but I thought they said over damped under damped first. So if it's less than zero it is under damp. So that should be that. No I had it ready I think. Yeah. So I see going down here it's definitely not right because I know this is uh she's anyway I think I had it right the first place. So if this discriminate is positive meaning that that means we'll have real roots of our characters equation. Which means that well if the system if it's a mechanical system we're going to assume that it's it's gonna be under damned and so we'll see that the roots all gonna be um all gonna be negative. Yeah so uh let's see here just cause you're gonna have a negative B. Plus or minus. You know something that is going to be less than B. So we have um this thing if it needs to be over damped we need right let me just do it on a fight. If it's over damped we need this to be real numbers but there's to be positive. So characteristic routes to be positive. So this needs to be real numbers. So this thing needs to be positive. So that means C. Needs to be less than four now for it to be under damp this needs to be negative because that you get complex roots of our character situation. So for that to be negative C needs to be greater than four. Yeah. Yeah. Yeah. I'm getting confused because I'm thinking C is the damping coefficient? And I'm thinking well if the damping coefficient goes up that should give more damping and more over quick. But see is our stiffness can see as our stiffness because okay as the stiffness goes up. Yeah then that makes sense. And then for critically damped we just know that this thing needs to be zero so that means C. Needs to be equal to for okay there we go. Um I taught vibrations for 20 years so when I see, see I just immediately think damping coefficient. So that's kind of why the kind of for the confusion here. I got to remember that. That's not what they're using here. So in 37 we have s double prime, it sounds too square to S prime plus C. S equals zero. Ok, so and again I don't remember seeing. Is this difference? No let's see here, discriminate is going to be B squared which is eight minus four C. And we need to find out whether that's greater than zero equal to zero, less than zero. So to be to be um over damped this thing needs to be less than zero or greater than zero. Sorry? So that means he has to be less than two. All right now to be over there to be under damped. This thing needs to be negative. So we have complex roots of our characteristic equation. That means he needs to be greater than two and for critically that that's just in between these two. So this is zero. And so that means he he calls to now for the next 1 38 is let's see here S double prime plus six S. Prime plus C. S. Equals zero. I'm going through the same thing to discriminate is 36 minus four C. Now whether that's positive negative or zero tells us whether it's over damned under damped, critically damped. Over damped means that this is positive, which means that see needs to be less than nine for over for under damp that needs to be negative. So we have complex roots of our characters equation. So that means he needs to be greater than nine and for critically damped it needs to be zero. So she needs to be nine then 39. We have let's see here. S double prime plus B. S. Prime plus five S. And so let's see here discriminate is b squared minus 20 is greater than equal to or less than zero. So the first thing they are over damped. This thing needs to be less than zero. Oh sorry, greater than zero. So we need to be to be greater than the square root of and I'm assuming be as positive. So you don't really ever have in mechanical systems and negative Supreme constant. Although theoretically you can construct something like that but not in a simple way. Um So I'm gonna assume be as positive. So B is greater than it needs to be greater than five times the square to to you could also saying it has to be less than minus five times square or two. But again that's not really. If we're talking mechanical systems which would usually are when we're talking over damped or under damp then B is going to be positive the spring constant. Now if further be, let's see under damped B needs to be less than a square of five times square to to and to be critically damped. He needs to be equal to five times the square to to and now in 40 we have, let's see here, 40 we had S double prime plus B. S. Crime minus 16 S equals zero. Well now all of a sudden we have this negative spring constant. So what do we have here? Um We have B squared minus four C. Is b squared plus 64. Now the problem is, is I'm not sure exactly how they're defining over damped, under damped, critically damped. If you say it's over damped damped usually means exponentially decaying, but because the stiffness is negative, these are all gonna be exponentially growing functions. So I would say that you can't find a damping coefficient that would make this critic over damped because over damp means exponentially decaying. Um Well exponentially growing means unstable, so I would say none. Um The other answer is possibly all if they're saying any exponentially type behavior is over damped, but again, that would be really poor definition of over damned. Now they asked for under damped, well you're never gonna get this is never going to be less than zero, so you're never gonna get complex roots of our character situation here. So um basically there's no values of B that we could make this um critically under damp and likewise we can't, there's no values, p no real values will be anyway, that makes this thing zero. So again, we had to have no values of the damping coefficient that could make this thing critically damped. So I'd say the answer to all of these is none.

Hello. We have to solve the given 3rd and 4th order differential equation. That's a dick. You buy 160 square by plus 11 divided- of six. by question zero. We can write the house. Every question that is MQ -6. Semi Square. That's 11. M -6. Close to zero mm. We can write it M -1 and -2 and M -3 -1. multiplied by N -2. and a monastery close to zero, swim one is one. M two is 2 and M three is three. So solution came in tunis even into Italy power of M. N. X. Bless you. To E. To the power of M. Two X. Plus C. Three. H. E. To the power amp and three X. So I will be seven. You do the politics plus ito into the power two weeks plus C. Three. It took the power of three X. So this is the answer. I hope you understood.


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