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10 . Find continous solution to the IVP dy 0 < € < 3 dx +2-{2 3 < :9(1) = 0...

Question

10 . Find continous solution to the IVP dy 0 < € < 3 dx +2-{2 3 < :9(1) = 0

10 . Find continous solution to the IVP dy 0 < € < 3 dx +2-{2 3 < : 9(1) = 0



Answers

Find the general solution to the given differential equation on the interval $(0, \infty).$ $$x^{2} y^{\prime \prime}+9 x y^{\prime}+15 y=0.$$

Hello and welcome to problem six of chapter three. Section seven he were asked to find the general solution to the given differential equation which is Y double prime plus nine Y equals nine seconds squared of three teeth. And to start this we'll find the solutions to the characteristic equation which is R squared plus nine equals zero. And this will simplify to R equals plus or minus three. I. So why one sign of three T and why two Is co sign of 3. 2. Great from here. The next step is to find the Ron skin because it's relevant to the solution for method of variation of parameters. When we're finding extra term to include this nine seconds squared of 32. So the run skin Is just to buy two determined of why one and that's why one, Y 2 and its derivatives. Um So Yeah that's right out. It's a sign of three T. three. Co sign of three T. Co sign of three T -3 sign of three T. Mhm. And this simplifies to right well negative three Sine squared three T -3. Cosine squared 32. Of course this simplifies to -3. Okay, so our own skin is negative three. And from here we can calculate the equation for our capital Y. Of T term which um via method of variation of parameters why is equal to c one Y one plus C two, Y two plus live to we're finding live to hear which is equal to Uh negative. Why one integral of Y two times G W. D. T. Where G is just this and we'll add this to uh why two times integral of why one times G. Over W. D. T. Of course. And we can just plug everything in. Since we know all the values will have uh negative sign. Three T. Temps integral Co sign of three T. Let's move that. Let's move this. Run skin to the outside. So have a plus 3rd on the outside. And we'll be multiplying by G. Which is nine um second squared three T. Well we can simplify this one more time to being um Times nine over cosine squared of three T. And we can add this to I don't know if we're gonna fit it in but it will be um Y. two Just co sign of three T Integral of why one sign of 32 times nine over. Cosine squared of three T. Can't forget that Run skin. So we'll have to subtract this divided by three. Oof that's tight. So let's try to quickly simplify. Get out of this. So from here we can cross off that. Co sign let's do it in red. Co sign and we'll just be left with one co sign. Um Here we can change this into a um So it looks like it's not going to simplify very well but if we need to be a second times tangent. So let's uh All right let's move this all down here. So we'll have Uh nine on top will turn into a three Y. Two. Of course three Sign of three T. Yeah Times The Integral of 2nd of three D. D. T. And subtract this from oh yeah three co sign of three T. Times. Well it looks like the integral of second three tee times tangent of three T. Second 32. There was a tangent 32 Gt fucking get that in there. Well we're almost done there but you have to find these two integral. And what is the integral of 2nd of three T. Well. Well it's um one third times the natural log of tangent of three T. Plus second of three T. Um The integral of sequences. One of those annoying ones that you should just memorize because it comes up enough. Um But driving this is as a pain so I will skip this. Um It will have uh this is equal to three sign of three T. Times Well there will be 1/3 in the solution to this. So cut that out safe space. We'll have sign three T. Times the natural log Of Tangent of three T. Plus, seeking to three T. And we'll subtract this from three. co sign of three T integral of this. Well what is integral that? Well that's one third second of three T. So um That's just 1/3 over 1/3. Co sign three T. So this is just going to simplify to one. So Cut that out to just be -1 minus one. Okay. And that is our Y of T term from here. We can write the final solution. We we expressed up here as just don't blue Why is equal to C1 Sign of T Plus C two. Co sign of T plus. That's actually should be sine and cosine of three to just three. Three plus Sign of three T times the natural log of tangent of 32 second of three t -1. And that concludes the problem.

Hello. Good of you. Problem number 53 from the section Doctors Review here we heard to find the resolution of the given different city question. Did you want definitive creation is and to square by double less plus nine X right? It does plus 16. Why equals zero Led by a little extra about our so it can let it burn us. I do. My Nesler plus my, uh, best 16 in Quincy, Ill. Bigger us minus are blessed. Mina plus 16 equals ill. Both square plus eight plus 16 equals. Bigot, huh? Breath for the was quite because our Nick was mindful. Come on. Mind of foot for the solution. Gonna be feeler existing minus four plus c Do existing minus four and let X That's didn't offer a question. Thank you.

Okay, So what we have here is a differential equation Working were asked to get it solutions you. Then its initial review showed, which is off white at zero is equal to 10. So what we want to do first this is is to transfer. Why? Over three to the left side, to the right side. I mean off the pressure. And this will give us the Y over the eggs. Physical negative 1/3. Why next? We removed it by one over. Why the both sides and we will have the wife over. Why the X is equal to negative 1/3. Next. I wonder what they liked the ex to both sides, and this will give us the way over. Why is it called a negative 1/3 X? So integrating both sides of the equation we'll have and in a way is equal to negative 1/3 X. Let's see. We can also express this a ski race to Ellen of why is equal to either use the negative 1/3 eggs. Let's see. So we know that here is the island of why it's just why and see it's just a constant so it can take any values. Therefore, we can express this as C E brings the negative 1/3 x So this is now our generals should So we want to find the particular solution when we do the speck of getting for the value of C. So in order to compete toe compute first see, we make use of the initial conditions given, which is at X is equal to zero. Why is equal to 10? Therefore we substitute this So the general solution we have then is equal to see erased a negative one over B zero. This will just be equal. The one and the value off C is equal to 10 rewriting the clinically shown we have. Why is it well then erased in the negative 1/3 eggs? And this is now our final answer, which is the particular solution off over differential equation.

Alright suffer from 11. We have to find the general solution to this differential equation. And when you have a simple differential equation like when all the coefficients are constants, then you can imagine that the solution is going to be in the form of E to the power for constant times, the independent variable. And I'm just gonna put X for the independent variable here. So we're gonna match in. That's of the solution that is going to be in this form. So now we dressed the differentiate. So why prime? That's gonna be K time. Thio Thio Park, Ajax Why double constant being K squared southeast of parquet acts. And now we substitute these into the into the differential equation we want to solve. So it's gonna be a K squared Planets eating to the car K x minus of three times k times easier dark a X minus, 10 times easier park Ajax goes to zero. We're going to factor out a k e to the power k X from all the terms. So it's gonna be each of the parquet acts times que score and minus three K minus 10 and see what his hero and Now we have toe. Make sure like when this part is gonna be equal to zero, since you haven't exponential function on the left, which will never equal to zero. So we can only rely on the second part being equal to zero and conveniently is Jessica for drug equation. So it's gonna be case growing on this three K minus 10 if he was zero, we're going to factor is, since this fact herbal, we're just going to give us K minus five times que my okay plus, which will give us a secure your secret and negative on five. And just like that, we found the solutions to our differential equation, which is gonna be lying sequel to a constant one times E to the power of negative two x plus. We're like, add the two together. So it's gonna be another constant times eat of car fire necks and yeah, that's basically adds


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