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Find all intervals on which the differential equations shown below are normal: (a) Vxy" +13xy' _ Iy= In(x? _100) (b) (I+x )y"+xy"-xyty=0 (c) x -...

Question

Find all intervals on which the differential equations shown below are normal: (a) Vxy" +13xy' _ Iy= In(x? _100) (b) (I+x )y"+xy"-xyty=0 (c) x -e2 (Ingxl - 2lne)y + (cschx)y = 0

Find all intervals on which the differential equations shown below are normal: (a) Vxy" +13xy' _ Iy= In(x? _100) (b) (I+x )y"+xy"-xyty=0 (c) x -e2 (Ingxl - 2lne)y + (cschx)y = 0



Answers

Match the graphs of solutions in Figure 11.108 with the differential equations below. (a) $x^{\prime \prime}+4 x=0$ (b) $x^{\prime \prime}-4 x=0$ (c) $x^{\prime \prime}-0.2 x^{\prime}+1.01 x=0$ (d) $x^{\prime \prime}+0.2 x^{\prime}+1.01 x=0$

In this problem we wish to solve to give a differential equation DVDt plus to be equal safety. For initial conditions, be of what equals 100. This question is telling your understanding of how to solve differential equations specifically via the separation of variable. That's it. Which requires three steps and step one we actually are being are two terms on either side of the equal sign and steps. Should we integrate both side of the equation and in step three we saw for the initial conditions, So proceeding a sec one we can rewrite D V D t equals 15 minutes to be STB over. Be managed 25 equal negative two DT. Thus we integrate both sides to obtain L N B minus 25 equals negative two T plus C. Exponentially Eating, adding in 25 and we're ready. You know the C. S A gives B equals eight years and I get to teach was 25. by plugging in our initial conditions in step three we can solve for a thus 100 equals 80 is negative two plus 25 or a equals 75 square plugging in. Give solution B equals 75 to the two minus two. T plus 25.

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 general generic differential equation. So we'll keep with their notation. So basically they asked 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 main 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 overcrowded. 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 crime times two squared 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 C equals two. 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 it's a 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 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 damped. 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 can make this um critically under damp and likewise we can't, there's no value to be, 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.

So we have the function. Okay? It goes to ax minus one over X. Quite so we can integrate both sides. Yeah, it goes to. And those one Jesse Helms Huy and down the rest that will have X choir over to thus one over X with constant. And also here we have a initial condition. That is when X equals to one y equals to two. So take that into our function will have. See here, Jessie close to ah Huff. So you have the function. Why equals two x choir over two plus one lower x plus a half. And for B, we'll have our initial condition. Why minus one equals to one. For this time, we'll have our constant here. Just equals 23 over to. So the new function for this one equals two x choir over two plus one over x Last three over to and access leiston zero

So this problem we want to find a solution to this differential equation. And to do so, we should integrate both sides. Um, the indefinite integral of Why? Double prime. We're just going to write as why Prime. And here we want to take the integral of X e to the x d x here. We need to use the integration product roll. So you DV equals u V minus the integral of the do you real set you equal toe X V equal to either X and playing. And we get that X either the ex, do you We're sorry. That would be d X equals u times v x times e to the x minus the integral of you the x dx. So in that case we get that are integral of X of the X DX is equal to x times easily x minus e to the X plus C one. So that's equal to y prime. And now we need Teoh find we need to integrate again to get why. So we get that why is equal to the indefinite integral of x times e to the X minus two the X plus c one some constant d x. So we get that why it's equal to we can break this up. We know we just did what the integral of x of the X is, and so we can go ahead and write that as ex either the X minus, eat to the X and then the integral of negativity to the X is minus eat of the X. The integral of C one is plus C one X and then we need another constant here. So plus C two and so we get our final answer. Why equal to um x Times e to the X minus two e to the x plus c one x plus seed to


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