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Consider the ODEy" + Bxy' +y = 0(1)Find a (constant) value of B so that at least one non-zero solution of (1) is just a (finite) polynomial, instead of an...

Question

Consider the ODEy" + Bxy' +y = 0(1)Find a (constant) value of B so that at least one non-zero solution of (1) is just a (finite) polynomial, instead of an infinite series. Actually, there are an infinite number of choices for this constant B that give finite, nonzero polynomial solutions. So to pin things down, choose the value of B that gives a polynomial of 3rd degree. Finally, to completely pin things down, make sure the coefficient of in your solution is -1 , Find this solution: Ch

Consider the ODE y" + Bxy' +y = 0 (1) Find a (constant) value of B so that at least one non-zero solution of (1) is just a (finite) polynomial, instead of an infinite series. Actually, there are an infinite number of choices for this constant B that give finite, nonzero polynomial solutions. So to pin things down, choose the value of B that gives a polynomial of 3rd degree. Finally, to completely pin things down, make sure the coefficient of in your solution is -1 , Find this solution: Check to make sure your proposed solution actually solves (1) for the indicated value of B _ What is B ?



Answers

Consider the differential equation $$x^{2} y^{\prime \prime}+x y^{\prime}+(1-x) y=0, \quad x>0$$ (a) Find the indicial equation, and show that the roots are $r=\pm i$ (b) Determine the first three terms in a complex valued Frobenius series solution to Equation $(11.4 .15)$ (c) Use the solution in (b) to determine two linearly independent real-valued solutions to Equation (11.4.15).

A second order. National questions. Why? Don't find minus wife run. I see one. He goes to one witness, X, we're gonna party. Great solution where you look, Dysfunction. That's where this one soles The's a question. Burn on defined the general solution. So this is your from the function of this one? So, uh, is a general So it's check first. That least satisfies the questions. So that is it true that people was mine is the first Mr Finish motion until people one Do you think this function there was there? So I'm just one on the second. You So we were there. Zero. Yeah. Weightless one. This is the Why does That was unfortunate till you'll explain was one, um is your vehicle to my house? One school minus minus. So plus two. So why us want Waas through one with the about these work? So this dash will be satisfied to solve these equations that we want to get this general question. You need to watch young with Jews. I'm a genius front. This is a fortune. Such that next, these equation you go to zero using the seniority. This so well Since we have the other serially affects marking its That still, despite the question, but is more general. So these are trading a bit off the Terran conditions for Houston. So where did you have, uh, r squared? Wait, so what do? Zero. So the solutions that are moving up to number that I wanted, like again Yeah. Minus still on r minus. Wants you would be up my nose through on one. Yeah, to these. So that young. Only because you wouldn't make standard that these flus are minor schools of these people minus one. So the general, you know, I'm a genius with the, uh, one explain int all these surgery for these articles position the society minus one. It's gonna be two x plus minus six. Plus, we'll on our close and he you did that to them miners X You didn't mind. Or six. So because there are more genius on the well to make it said, despite so general solution is energy. Why, It's gonna be OK. That was last B minus six. Was that for people about six minus one. So we have this function, so the function is not having they off. The function will be really friendship is you get to a I didn't do it. Well, these minor today my sex on their loved one So that, um Well, you can we have blood in these data living in this data change any tappy spice to to solve these constants? A Me. So what is gonna be these equation that's you will be well, just burning there. Yeah, serial, That is what must be for this year. Bless you. Minus one. So that has to be able to last people is you on the regrettable conscionable With that zero would be discuss. It will be again. So in this year Oh, my house be tempted to this year on my scream plus one So that ask gave or toe one on what we can You cannot These two equations So we a long things will cancel Yeah, them So the situation waas five mission These one most That equation will be will be that on the side You get a bust way. Okay, was the my week Mine is wormed plus one that has to be equal to syrup was one So how's biblical one? So 111 seal being envies you and then This is 381 So from these, we can solve that. A one there. And so I'm replying into these equations that you know, one head. He's a minus one. So Well, Morgan, he's there about there. Have you been to one? Well, we need their changes. Sign 41 final on Juan Ramos of Third. The same less be they're explains 1/3 to so stay. Embarrass me. So the general solution would be gays. Why be a so 1/3 today? Two x plus you thirds. So you 2/3. You didn't mind of six? Um, I lost six minus one plus. What? This is there general solution? Well, the solution That is fine. Police recent usual there exit mentions that he's fine, that they is the

Were given a differential equation in part A. We rest to show that the initial equation associated with this differential equation has exactly one route and to find the for venous serious solution for this route, the differential equation is four x squared y doble prime minus four X squared y prime plus one plus two X y equals zero. To find the additional equation for this differential equation, let's put this differential equation in standard form for a differential equation with singular value at X equals zero. So we want the coefficient of y doble. Prime to be X squared is divide both sides by four yet x weird y double prime minus x times X y prime plus 1/4 plus 1/2 x Why equals zero? So now our equation is in standard form. So we obtain the two functions P of X, which is the coefficient of y prime divided by X or negative X, and a queue of X, which is the coefficient of why just 1/4 plus 1/2 X see that both P and Q are analytic at X equals zero, which implies that this equation has a regular singular point at X equals zero and we have McLaurin, Siri's for P and Q are valid for all real numbers since Corrine Siri's there. Simply finite pollen, No mules. So the initial equation is given by our times ar minus one plus p zero, where P zero is the first term in the MacLaurin series for P, which is simply zero R plus Q zero, which is the first term in MacLaurin series for Q or Q of zero, which is 1/4 equals zero. So we get r squared. You want to play there by four. Four r squared minus for our plus one equals zero or factoring to our minus one squared equals zero, which implies that where is going to be equal to 1/2. So we see that this equation only has one route are 1/2 bunts by a term in the book, we know that there exists for being a serious solution to this equation. We know that this solution is going to be of the form. Why of X equals some from an equal zero to infinity of a n x to the n plus r. This implies that why prime of X is going to be some from n equals zero to infinity, differentiating each turn the some we get. N plus R. Times eight n x to the n plus R minus one and then y doble prime is going to be some from n equals zero to infinity again, differentiating term by term. N plus r Times, n plus R minus one a and x to the n plus R minus two Since why is a solution to our differential equation, we have the four x squared y doble prime, which is one of the four times n plus R en plus or minus one a N times, Ecstasy and plus R minus two plus two or M plus r from n equals zero to infinity minus four x squared y prime. This is minus some from n equals zero to infinity of four times and plus R A and X to the n plus or minus one plus two or in plus are plus one plus one plus two x y. This is actually two different sums, so we have plus why, which is simply the some from an equal zero to infinity of a seven x to the n plus R and then two x y is plus the some from an equal zero to infinity of a n or 2 a.m. X to the n plus R plus one equals zero. Dividing both sides by X to the are and combining like terms. We get some from an equal zero to infinity of four times end plus R Times end plus R minus one plus one a seven x to the end and then from that, subtract substituting in sums to and four and minus one for end the some from any quotes. One to infinity of four times and minus one plus R minus two times a n minus one x to the n equals zero. So on one side we have power. Siri's among other side. We have zero. This implies that every quote fishing to the power Siri's must be zero, and so we have in particular, and is equal to zero. Four are times ar minus one plus one equals zero. This is the same as the indicia equations doesn't tell us anything new about her solution. If an is greater than or equal to one, however, we have that four times and plus R Times and plus R minus one plus one times a n minus four times and minus one plus O R, minus two a n minus one equals zero. We know that are gonna be 1/2 so it's substituted or equals 1/2. We get four times n plus 1/2 times and plus 1/2 of minus one is and minus 1/2 plus one times a n minus four times and minus one plus 1/2 is and minus 1/2 minus two a n minus 10 We have the n plus 1/2 times in minus 1/2 and squared minus of fourth. Times four. We get four and squared minus of fourth times four, which is one plus one simply foreign squared a n minus and then for end minus two minus two is four and minus four. So we get minus four times and minus one a n minus one equals zero. Or that a M is equal to four times n minus one A and minus one over four and squared so we can cancel out the floors and get over and squared mrs. For n greater than one. So we have in particular, a one is going equal to zero times a zero over one squared is going to be zero, and we see that in fact, a N is going to be equal to zero. We're all in greater than or equal to one. So if we take a zero being one, we have that intravenous. Siri's solution. Call it why one of X. He's one equal to X to the are, which was 1/2 times one. So we have solution. Why one of X equals x 2 1/2 And we know from the theory and this is going to be valid for all positive real numbers. This is the solution to part A For part B. We were asked to find a second linearly independent solution on the positive real numbers using reduction of order. Nordea's reduction of order Let why one of X beer solution party so x to the 1/2. You know that why one of X is going to be a solution to our differential equation, which we can write our differential equation in the form. Why Double prime minus y prime plus and then one plus two x over four x squared y equals zero creamery, right as 1/4 X squared Plus no 1/2 x no, no. So that the coefficient of y doble prime is one. We know this is a solution on the positive real numbers. Then we had from a the're, um in the book that if we take why of X to be equal to you of X, why one of X for some function you of x differential? If we substitute this function into our differential equation, then we will obtain the general solution. So and this will be on all the positive real numbers, huh? So in particular we have that. Why Double prime index or it's differentiating Turn by turn me of why Prime Vex. It's going to be by the product world for derivatives. You prime of X. Why one of X plus you have X. Why one prime index and then why Double prime of X is going to be equal to I began by the product rule you double prime of X. Why one of X plus and then we have you prime of X y one printed X plus you prime vex. Why, when prime index surplus to you One prime index times. Why one prime index plus you wanna you x? Why one double prime fix. So plugging this into our formula we get not right this different order you have X. Why one double prime of X plus to why one prime of X You won find of X plus Why one x you double prime of X. This is why double prime minus y prime so minus again in a different order we have Why one prime of X? You have X plus Why one of X, you have X and finally we have plus 1/4 X squared plus 1/2 X. Why times why one x you X equals zero Now we can rewrite this as you of x times. Why wonderful prime of X minus Why one prime of X plus went over four x squared plus one of her two x times Why I wanted X and then add to that you prime of X. So we have to Why one prime minus Why one and plus you of x times Are you double private x times why one X is equal to zero and now noticed that this part here is simply or differential equation with why one plugged in which we know is a solution This is going to reduce to. And if we divide by, why one X we get you double prime of X plus to why one print vex over why one of X minus one times you prime of X equals zero because we know that this expression and red is equal to zero. So we've used reduction of order to change this into if we said w of X equal to you Prime of X, this is now an equation with were to or order warning you. So this is going to be the BU prime next plus to why one prime of X over why one of X minus one and then you prime is simply w of X equals zero. So plugging in for white one of X, we have that w prime index plus why one print Becks is going to be 1/2 X to the negative 1/2. So we have when half times two is one extra negative half over extra 1/2 just going to be simply one of her ex minus one times W X equals zero And there are a few different ways we could solve this So as a separable equation could write this as do you W d X equals one minus one of rex times w of x so that we get that you know you over w equals one minus one of Rex E. X Integrating both sides we get in the girl dw w equals the integral one minus one over x gxe so on the left side will get integral of one of her. W is natural law, but w on the right side we have integral of one. It's simply X minus into grill. One of her ex is going to be natural. Law connects then plus some constant so that we have that w is going to be equal to e to the X minus natural of X plus c. We can use power loss to write this as e to the X and then either negative natural of x times one over x times eat of the sea and we can right you the sea as some constant quote See one. So have C one times e to the X over X so this tells us this is W you is the integral of w With respect to X, this is going to be the integral e The X over X were given a hint that in order to evaluate the integral of the X over X, we should expand E d x using its McLaurin Siri's I'm running out of space here, so I'm going to erase a little bit. So we have that you of X is going to be equal to the integral of W vex DX, which is see one times the integral of e the X over X dx. Now recall that ebx a corn Siri's is the some from n equals zero to infinity uh, X to the end Over in Victoria, that means if we're using the cornice McLaurin expansion, this is going to be the same as see one times the integral of this, some from an equal zero to infinity uh, ecstasy and minus one over in Victoria, the X and we can exchange the integral and the sum to get C one some from n equals zero to infinity of the integral, uh, extra. The in fact, I'll pull out the and factorial as well. So one of her, in pictorial times the integral of X to the n minus one DX I'm gonna clear a little more space here again. This is equal to see one times the sum from and equal zero to infinity one over in factorial in the integral of X to the n minus one we have is going to be X to the and minus one plus one ex to the end over. And as long as N is not equal to zero so and is equal to zero we're going to have to write this little bit different like this is the some from n equals one insanity of one of her in factorial x t end over end. This is a parentheses. It's a little bit sloppy looking Plus And now we have 1/0 factorial times the integral of X to the negative first. So whenever X would just simply natural log of X and this is gonna be all plus some constant call, it can see it will change pretty quickly. This will be equal to see one times this some from n equals one to infinity, uh, one over. In times and factorial X to the end plus see one times the natural log index plus and then see one time see will call this seat to in a race a little bit more to make over more space. And so we have that. The full solution to this equation is why of X equals Why one x times you x. It's going to be X to the 1/2 times. Then we have C one some from and equals one to infinity of X to the end over n times n factorial plus natural log of X plus C two so that this is the same as see one x to the 1/2. We can actually add this inside some to get C one the some from and equals one to infinity of X to the n plus 1/2 over end times in factorial plus x the 1/2 times natural log of X plus C two times X to the 1/2. So we see that if we take C one to be zero and C to be one, we obtain our function. Why one again in the racist? A little bit more almost done so in order to pick a linearly independent solution to this differential equation. But want to pick, for example, takes see 12 B one and C two b zero. So if we take C one equals one, C two equals zero, we get that we have a second nearly independent solution. Why two of X, which is going to be X to the 1/2 natural log of X plus some from n equals one to infinity of X to the n plus 1/2 over end times in factorial. So this is our second linearly independent solution, which we know from the serum in the book on a reduction of order, is valid on all positive real numbers.

No. Please question. So checking their give off water. It was the first she can order second order differential. Uh, question are being with even that the particular solution she called the O six square house minus X With these way, were you more really getting that? The solution, abs, you has to be you. The solution of all the ways that the results was here. So all the most general solution in this equation, he's gonna have me. I know, genius. A vision plus the articular service, uh, function on the on Virginia's all these since the question is cleaner and my genius solutions. So this fire So his equation, the situation being able to do, um, so by the matter off on the German quick visions, we assume that dissolution is even. There are tics. Oh, because thank for communities of these being, uh, no for use equation corresponding. Pulling No, no. Would be r squared. Lost our I hope you also there. No, it's for young community. Our times are plus one cereals of that has solutions. One submissions are people zero Another one. Is these being able to our 10 So things that are people who might as one. What we've in mind. Wander So speaking, lines one. So that means that there no more genius. So don't you? Does your niece love some constant? Just games issue. Plus, he I'm seeing bad number. I'm sex of my music. So the all the solutions one being able to now what genius? I need my specs. Was the article. Thanks. A squirt minus X. And so we'll, uh you can figure out what our discourse, since by that why must satisfy these collisions? Have you ever, Lloyd? Why at zero these Y oh, why people toe a plus the comes to this year's off. Plus, no, last year's were my cereal, so just the people who is you. So you have the questions that, uh, plus, he has two vehicles here. Then they think off dysfunction. Jowl be a My house mine Just married. Explained there minus no beautiful minus six. We're differentiating that plus fool eggs do my one. And so it is gonna be that for But while these expression X that X equals zero on this expression, Alexis, it becomes my most be today. Meet them seek with zero. So my team serum Maier's wife. So, uh, well, we have these questions for on these severity has zero chance of a zero. So we have that. We must be the people. Zero. Uh, hey, my b mind those one. You see? What was you? So we moved they this year. So you know, any secrets, my nephew and then these over here by replacing these would be a day plus a like this one about their get well, I'm having this till we get to a people to So we know from these a a people to one hot. I'm the C minus eight. I'm looking by this one. So peaceable. So we know that a c minus one out. So that would give us of the general solution unease. It's find those conditions is gonna be, um, 1/2 one, huh? One day. Um, the sequel to minus a some minus my boss. I need to get my saves house. Plus Tom exes for huffs. Um, my six. So set up these with the sotherby's. Um all that is that this is there. This solution use is that this illusion is fine. So this fire this the commission? So, Chris Fine. That I'm both very financial question


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