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L3HLUce the Frcbeniua mechod to Ed &ll tzns of the series xepreseutation around zo = 0 for the two lireariy independent solutions 9(2) and i(z} for the equation...

Question

L3HLUce the Frcbeniua mechod to Ed &ll tzns of the series xepreseutation around zo = 0 for the two lireariy independent solutions 9(2) and i(z} for the equation: 4ry" (s) 2(-)+ylk) = 0

L3HL Uce the Frcbeniua mechod to Ed &ll tzns of the series xepreseutation around zo = 0 for the two lireariy independent solutions 9(2) and i(z} for the equation: 4ry" (s) 2(-)+ylk) = 0



Answers

Show that the indicial equation of the given differential equation has distinct roots that do not differ by an integer and find two linearly independent Frobenius series solutions on $(0, \infty)$. $$2 x y^{\prime \prime}+y^{\prime}-2 x y=0$$

Were given a differential equation and were asked to show that the initial equation of this differential equation as distinct roots, not different. Buy into your and to find two linearly, depending venous serious solutions for this equation. The equation is six x squared y doble prime plus x times one plus 18 x y prime plus one plus 12 x Why equals zero now? In order to find the conditional equation for this differential equation, we want to write differential equation in standard form. So I want to divide through by six. So we get X squared y double prime plus X times one plus 18 x over six. Why problem? Plus one plus 12 x over six. Why people zero. And so now this is in standard form we can obtain from this function p of x, which is the coefficient of y prime divided by X. So this is going to be one plus 18 x over six or 16 plus three x and the function Q of X, which is a coefficient of why just 16 plus two s and we have that the initial equation is going to be our times ar minus one was he hit zero or plus Q of zero equals zero. And so it's becomes our times ar minus one plus he has zero is just six or plus Q of zero is the sixth equals zero. And so if you multiply through by six on both sides, we get six are times or minus one plus are plus one equals zero. So we get that six R squared minus six r plus r is minus five are plus one. It was zero. This could be factored as three are to our then minus one nice ones three R minus one times two on minus one equals zero so that the roots of our additional equation or or equals 1/3 and 1/2. So first of all, we noticed that the roots of the initial Beijing are unique. They're really members, and we take our one to be the greater of the two so 1/2 and are to be lesser of the two. Third difference between r one and R two is not an integer, So these are the distinct. It's not differing buying into your next. We want to find two linearly independent for being a serious solutions, so I have fear him in the book. Since we have that dysfunctions p and Q. Our analytic at X equals zero we have read the two routes of the initial equation are distinct and non into your difference. It follows that this equation as to solutions which are four yea, serious solutions. And we have that these two solutions are going to be linearly independent on the interval which they exist. So first of all, it's determined on what interval these solutions exist. We have that these serious expansions for P of X in q of X at X equals zero is defined for all X. And so we have that If we think of the opening terrible on which P and cured both find this is going to be all of real numbers, negative do infinity. And so we have by this, the're, um that, uh, solutions will be valid Forex from zero to positive and to find the solutions say one of them is why of X. This is going to be some from and equal zero to infinity of a seven x to the in plus or then it follows that why prime of X I was going to be something n equals zero to infinity of n plus r times a seven x to the n plus r minus one. And that's why double prime of X is some from n equals zero to infinity of n plus r times and plus or minus one. These have been extra VN plus R minus two, and we know that the solution is going to satisfy our original equation. Six. Expert. Inaudible Prime plus so on Equal zero. So we have that six experience. Likable crime. She's going to be the son. An equal zero to infinity of six times n Plus r times and plus or minus one ace have been X to the n plus r and then plus X times one plus 18 x y prime. So this is actually going to be two different life crimes. So we have X times. Why prime, this is going to be some from n equals zero to infinity. Uh, and then wife crime is n plus r. These have been the next to the impulses are and then we add to that 18 x squared y crime. This is going to be plus the sun from an equal zero to infinity of 18 en Plus or ace have been X to the n plus are plus and finally we had to that one plus 12 x y, which is going to be another two sums. So we add why, which is the sun from any zero to Infinity Basie band next PM and 12 x y, which is something any go zero to infinity of 12 a seven x to the n plus one equals. So if we divide, both sides should be in place for by X to the are we get some from n equals zero to infinity off six and plus or n plus R minus one plus and plus or plus one a seven x to the end and then plus or the third some. We're going to substitute and minus one for ends, so we get the some from n equals one to infinity off 18 and plus R minus one. A sad end minus one x to the in minus one plus plus warm just next to the end and likewise, for the fifth turn fits some we'll substitute and minus one for ends gets well. One B plus 12 times Ace of End minus one x to the n equals zero. It's what we have is that the coefficients of this infinite point no meal must all be zero. And so, for example, and is equal to zero we have at six times zero plus are just going to be six or R minus one plus are plus one has people. Zero. We see that this is the same as our initial equation. It doesn't tell us anything new about our solutions, but consider when and is going to be greater than or equal to one. No. Then we have that six times in plus or times n plus R minus one WAAS in plus or plus one times a seven plus 18 times n plus R minus one plus 12 times a suburban minus one equals zero. This could be simplified to Yeah, well, let's pull you in the value of our first. So first we plug in our one. We determined what is 1/2. So take RTV 1/2 in. This becomes six times and plus on half times and minus 1/2 plus and plus 1/2 plus one he's have been and then plus 18 and minus 1/2. I was 12 times he's been minus one equals zero. It's gonna be written as we have six times the difference of squares, So end squared minus fourth plus n plus three halves times a seven los and then 18 n minus 1/2 that's 18 then minus nine. Plus 12 is three. It's been like one equals zero with the plane, both sides by four. Or instead expanding this out. We get six and squared, minus three Hades plus three halves plus ends six and squared. Plus in times a seven plus three times six n plus one. It's a then minus one equals zero, and so it is written as a seven equals negative three times six and plus one. These events like this one over end times six and a close one 67 plus ones canceled out. Simply give us negative three a sit n minus one over em. And so we take end to be one. For example, we get that a subordinate to maybe three. A zero over one. If we take end to be to Ace of two is going to be negative. Three. A one over two, which is the same news Positive. A zero three swear a zero over one times two take end equal to three. Get at a three will be equal to negatives. 382 over three. We want people to the opposite of real cute, you know, over three factorial and so it's clear the pattern to be approved by induction a N is going to be negative. One to the end times three to the end, not over. In fact, where N is greater than or equal. So we have that one of our for being a serious solutions. It's going to be why one of X which if we take a not equal one this is gonna be X to the Nicole that we choose Arditti 1/2 extra 1/2 times one plus some m equals one to infinity Negative one to the end Agree to the end over in factorial And this can also be written as X to the 1/2 times some from n equals zero to infinity of negative one to the end not just negative ones the end, but negative three to the end Over in factorial we recognize that this is the same as Times X to the end. Of course. Negative three negative three x to the end over in factorial, which is gonna be equal to excellent half times e the negative the re X. So this is one of our for Venus. Serious solutions written more simply find the other for be necessary solution. We'll adapt calculation so that instead of r equals 1/2 we'll make our two, which is 1/3. So now we have or equals and we're going to replace all. The ace of ends with BC Events is the distinct coefficients. So our becomes 1 30 We have six times n plus 1/3 times in minus 2/3 plus and plus 1/3 lost one bn plus 18 n minus 2/3 plus 12 bn minus 20 0 So this becomes and squared minus 2/3 and plus 1/3 end is one B minus, 1/3 and minus two nights. So six times in squared, minus 1/3 in minus two nights plus and plus 4/3 he in plus 18 n in 18 times negative 2/3 to be negative. 12 plus 12 plus zero. This is gonna be n and yet not this one equals zero. So we get six and squared minus two in plus, and it's going to be six and square Linus and and then minus six. There is R minus 4/3 plus 4/3 0 So we have six months Weird Minds and Bien plus. And then we get 18 N e n minus one equals zero. And so we can write this. As he said, Ben is one equal to negative 18 the n minus one over six cents Word minus end is end times six n minus one. This time, we don't get to divide out any factors except for the end Says becomes negative 18 8 bn minus one over six and minus one. So, for example, to take and be one you get that E one is equal to negative. 18 zero over six times 16 minus one is five and equals two. Yeah, that B two is equal to negative. 18 B one over six times to his 12 minus one is 11. So we get this is then equal to 18 squared e not over five times 11 and then if n equals three, we have that be three is equal to negative. 18. You too, Over there. Its expense three is 18 minus 1 17 which is equal to negative 18 huge. You know what? Over five times? Seven times 11 with five times 11 times 17. And so we see that in general, a sedan is going to be negative. One to the end, 18 PM You know, over five times 11 times I'm six in minus one where and is greater than or equal to one. And so our second solution Why two of X is going to be if we take easier to be one. We have X to the 1/3 times one plus some from and equals one to infinity. A negative one The end section right as negative. 18 x again. And we're not just one over 5 10 11 times six and nine. And unlike with previous solution, it's not clear how we could write this solution or simply so. These are two solutions We know by the serum that these two solutions are linearly independent and are valid from zero to infinity. This is our answer

Were given a differential equation and were asked to find the roots of this different of the initial equation of this differential. Beijing To show that the roots are distinct, I do not find in future and finally to find two linearly independent for Venus. Serious solutions for this difference situation. The equation is X squared y doble prime plus X Y prime minus to plus X Y equals zero. We had that. The equation is already in standard form, so we get function. P of X, which is co vision of like crime about it by X, is simply one in the function. Q. Of X is the coefficient. Why just negative two plus X in the indicia LEP, Asian is our times R minus one plus he is zero plus q of zero equals zero. So we have that a war Times ar minus one plus or minus two zero so R squared minus R plus R minus two equals zero so that our is going to be equal to ulcer minus root, too. So we have that the roots are distinct and that's we are ones with greater routes. Just to are to be lesser, just negative too. We have a difference between these or when Minus are too going to be to To which is not uninjured. So you've shown the roots of the initial clean are distinct and during different. Therefore, we have a serum. What says that since our equation differential equation is in standard form and we have that are functions peeing que our analytic at X equals zero. In the roots of the initial equation are the state and you're not different by an intruder. We have that since arches, Syrians expansions for P of X and two of X. We're going to be valid for all X, so exit negative infinity to positive. It follows from this theorem that are difference. The equation has to linearly independent, serious solutions that are valid at least for zero less than X lesson. So but this there may have to linearly dependent for genius, serious solutions. Sure, that would 02 infinity and are nearly independent. To find these solutions is that why is going to be one of the solutions we know is going to have the form Bligh of X equals X to the are times some from n equals zero to infinity a seven x to the n, which is equal to the sum from any pool zero to infinity UH, a seven x to be and plus R. We have that. Why prime of X is then going to be some for men equals zero to infinity and plus or times hey sedan times X to the n plus R minus one and finally wind up with prime equal to some from n equals zero to infinity of n plus or times and plus or minus one and a sedan, times extra fee and waas or minus two. And so, taking this a solution to our differential equation, we have that X squared, likable prime. It's going to be some and equal zero to infinity of impulse or two minutes in us are nice. One. Wednesday's been X to the n plus or plus X Y crime. This is a plus. Something any go zero to infinity, uh, and plus or again ecstasy and plus R and minus to plus extends. Why? So we actually have two different sense here, minus some for medical zero to infinity to why I just want to ace of n next to the and thus, or and minus X Y just going to be the son n equals zero to infinity Uh, a seven. Next to the end, less are less well equals zero dividing by XDR And then in the fourth son substituting and minus one for N we get This is equal to this some from an equal zero to infinity, uh, and was or times n plus or minus one. Let's send us our mine's too times piece of been X to the end, minus some from and equals one to infinity, a piece of end minus one x to the and minus Wonderful circus one or M plus and my insulin plus one just XP in mrs still equal to zero. So we have that situs infinite. Pulling a meal is equal to zero. Coefficients must be zero. So we have that, for example, and is equal to zero. We get that in plus or times and plus or minus one plus and plus R minus two. You can place the ends zeros, so we get that our times Army's one plus or minus two a zero. It's going to be equal to zero. You know, the easier is not equal to zero. So it follows that this is simply zero. But this is the same as our initial equation. It doesn't tell us anything new. However. I suppose that ends when he created a new one and we have It's in plus or tons and plus or minus one plus en plus or minus two. He's been minus base have been minus one equals zero. You can write this as en plus or affected him times and plus or minus three times have been minus a sit n minus 10 And now suppose that R is equal to you're too, and we have. That's incident is going to be equal to hey seven minus one over and plus two times and plus two minus three actually stayed here so they should be and plus or times and plus or minus one plus one, which is going to be in plus or squared minus two times a seven minus a spangly one sierra. And so this could be factored further as a difference of squares. So this is actually the same is and plus or minus for two ties in us less room to times have been minus seven minus 10 You have that. If R is equal to two and we get that any times we get that have been is equal to a seven minus one over end times en plus two two. So, for example, we have that and equals one. Hey, Suborn is going to It's a zero over one times one plus 22 if any is equal to that 8 to 10 people to ace of 1/2 times two plus to to this is going to be equal to It's a subzero over two times one times, then we have two plus 22 times one plus 22 and is equal to three. We have a three people to case of two over three times three plus 22 just an equal to a subzero over the refectory. ALS times three plus 22 sometimes too close to to times one plus 22 So in general, this could be proven by induction, but we have that ace of them is going to be equal to, in short, over and sectorial times one plus to to times two plus to to coins and plus two two. So if you take a not to be one be attained. One for Venus. Serious solution. Why one of X equals So the X t or 12 times one plus then some from n equals one. Infinity uh, ace of end, which is going to be one over in factorial one plus 22 times two plus 22 all the way up to end close to two x to the end. So this is one of our serious solutions. Find another solution. We'll use the same calculation except instead of using one. Let's use our to this time. So now instead of our being positive too, it's make or to be negatively to. Then we have a spin is equal to case of n minus one divided by And then this becomes any times and Linus through the tube. We want change A's to B's. So he's been He spent much one of her end times n minus two or two. So NZ one you have that he's a born is equal to B C 0/1 minus to to and it's equal to we had a piece of to is equal to he said one over. Two times two minus to to or he's a 0/2 times one times two minus two to one minus to to. And so we see that, as in the case with AIDS, the general form will be, he said. Ben is equal to, he said, zero over in factorial times when minus to to thanks to minus to to times all the way up to in minus two to and so we have if we take, you know, to be done. Second, serious solution. Just linearly. Independent first. That's why two of X be extra er to, which is negative, too. Times one plus son n equals one to infinity uh, X to the end over. In fact, times one minus two to tu minus to t all the way up to and minus two. So this is the second Siri's solution, and this is what we wanted to find

Were given a differential equation were asked to show that the roots of the conditional equation associating with this differential equation are distinct and do not differ buying and then to find to linearly independent. For venous Siri's solutions to this differential equation, which are valid when all positive real numbers differential equation is two X squared wide of the crime, minus extent one plus two X y crime, plus two times for X minus one wide people zero. Find the additional equation before this different integration. That's right, this differential weighting standard form for a differential equation with singular x way he condition in light of a prime two x squared. So I'm going to like both sides and they get X squared. Why did you trying minus x times? One was to mix with two. Why pride? Once four X minus one. Why zero? And now that they're winning standard for extract two Functions P of X, which is co vision of white crime divided X negative 1/2 latest X and Hugh of X itches coefficient of y forex Might both of these functions our analytic at X equals zero and McLorin expansions our valley for all real numbers. The additional weight secretions are times plus p of zero R plus. Q zero equals zero for our times. AR minus one waas hit zero So minus plus Q of zero is negative. One minus one equals zero. Pulling this out, get R squared. Yeah, minus all remind us. A. Half are so minus three halves or minus one. Who's here playing with us? I'd like to get to R Squared three minus to Europe, which we can factor is to or or I want to have minus here steer and then minus two to R plus one times ar minus two. Checking me in two squared minus four are plus one or this negative three are nice to this is equal to zero valid factory ization. And so we obtain the 2/3 of the initial patient in order of magnitude. Greatest one first are one is equal to to or two is equal to negative. The roots are distinct and clearly do not differ. Buying into this guy here in the book. All those that since functions your analytical X zero and learn Siri's are defined. X and Richard in visual equation are students do not differ by Richard. There are two distinct for a Siri's solutions are different. Better now, X, and they will be linearly. Independent. No, that's it. So we're defined to consider the general for medias Siri's solution. Why vaccine Police Next or same from zero to infinity of a seven x equal to the sum n equals zero to Infinity event. Next. Therefore, you have a wide crying necks. It's going to be some any zero interview plus four times a seven ext end plus or minus. We have the boy from the decks syrup infinity times in plus minus one, a seven ext and plus or minus two. So we know this function. Y is satisfied to French clean. We have two X squared wide of a crime. Just zero and this or one x n plus r nights, too most U minus. X times one plus two x Like crime. It's gonna be trooping sons First we have minus X. Why prime minus some of zero? Uh, why prime in just or incident times X to the n plus or minus one plus X equals or then we had minus two x squared like crime is your community two times in this I'm next in our minds from coast to coast, plus two times for X minus one. Why this is going to win sums. First we have eight x y So we have plus some equals zero to infinity eight a seven extra V impulse are and then subtracting Negative two y minus some. Any zero to committee to ace event Next equals zero. This is their differential patient with the solution. And so we combine like terms. Get something from any zero to infinity two times and minus and plus or minus two a n Next to the end, we divide by XDR on both sides and then in the third and fourth, some will substitutes and minus one for N he gets plus the sun and one to infinity in one less are Linus. We have a sit n minus one and x to the n plus one minus 16 is equal to zero. So we now have on one side and polynomial either said we have these Europol. So this implies that all the coefficients of the infinite Paul McGreal zero. So this implies, for example and zero we have that to times are times our money This one minus R minus 20 You see that this is the same as the official. This isn't telling you about solutions don't are. You know, I have visions two times in us or plus R minus minus sport Venus too. 87 minus two times n minus one plus R Thanks, eight times a C and minus one equals. We can simplify this slightly. So intern factory of implicit more to get in terms really have two times and this or minus one. Just get this two times in, thus two times over, minus two minus two and then, in fact, the routes to minus two times and minus one plus R minus four en plus or minus. It's about minus one equals, do they? And we take great in a vision because and close to two and plus four months threes minus two times a seven minus two times and plus tu minus five was minus. It's a then minus 10 willing out first coefficient get two squared plus in foreign plus two reasons to get to in squared plus five event minus two times. And my, it's been my one Syria so we can write. This is equal to two terms and rings n minus one and tiles to end less. Five. So, for example, we have that it's one 1 to 2 times when the minus three is negative. Two times you know it's over one times two times two plus 57 we have that to a to be equal to two times two minus three is negative. Woman times a one over two times two times two is four was flying, I think, 1 to 2 times negative. So we have a two squared times negative one times negative, too. Your point over one times two times seven times nine. We have that NZ three, a three way girl in greater Than or equal ace of Ben's zero. So going that calculating me have that a warning is going to be negative. 4897 A two is going to be negatives. 19 a note. A one so negative wondering times May 7th for 63. So we get it for me. This seriously shoot for our equals to unity. Why won't fix he was next to the origin is X squared times. If we take a not people one get one. And it, Linus, for sentence X and plus for 63rd ex weird B squared, minus 4/7. Execute plus extra food. This is one of our serious solutions to find the other serious solution will use the same calculations except instead of RG two Now all the people to our to which was negative on hand. This tells us that in our equation becomes and minus 1/2 times to end minus one minus three to end lice for minus two A seven minus two times and minus 1/2 minus five. Please end minus 11 hands and for the coefficient base of and in fact, around two from two and minus. For the first factor to get two in minus one times and minus two minus two, he said then and then the coefficient of a seven lights warn you can buy to factor in mites, leading it to to get yes to in minus. Let me and we're gonna change needs to bees still wish provision. Siri's from each other way before allow coefficient to in squared my use for any lines. Points plus two minus +20 times. Yeah, minus. He's been two and minus 11. A and minus room equals zero. You write this two in minus 11 the memories over end times two. So with NZ one, yes. Do you remember the equal to turns? Living is negative times zero over one time, two times when it's two minutes 53 and it's equal to people to two times. Two is 49 11 7 reason. Seven times even over two times Putin's to this four minus five negative is equal to negative knowing times. Negative. Seven zero over one times two kinds. Negative. Three times negative one with three 32 Does three is six. Minus 11 is negative five to three times. Give 365 Positive one equal to negative. Nine times negative. Seven Negative, you know, over one times. Two times three lines Negative. Three times negative one. So it seems like in general we have that right in the way of space. He's a bit equal to negative. Nervous. Seven. That way. Have to tu minus 11 you know, over in factorial Negative three negative warm all the way up to do n minus five. So the question becomes coming right this more super for well and using. So this is for any end. Great. Ordinary one. Whoever we have, that one is 20 minutes. 11 would be greater than or equal to two. And three of the 12 inbreeding. Six. So we have that. Yes, six. A negative times. It's negative. I agree. Negative one. I give him 6 to 12 minutes. Up to now we have a warn off to one times three Weaver, in fact, Times negative. Three times negative. Three way up to two minus. So notice that twin minus 11. 20. That scene two times minus five minus one. And if you re right, that also the number here, two in minus five is the same. Two times minus two minus one. Would you rate this is too to end six. Let's one. And to end minus lead. That's mean. It's from one to two times in minus six. One. This is the same is two times in minus six. Plus one that for you, over product to four. Hopefully have to two times six. Compact. You're out? Yeah, and minus six twos from bottom to get two times in 16 plus one. That story over to the n minus six times one tends to six simply 10 minus six Factory. But it comes two in minus 12. Plus, we have that likewise product from one numbers. Two in minus 31 que to Henry's three plus to three times and my three plus one factorial. Just two men minus 6 to 5. Factorial over to be in minus three, minus really factory. And so if you take the ratio of these two, but get well, yeah, two and minus 11 factorial over two in minus five. Factorial that will give us to and minus four to end. You know you have 24 23 all the way up to to end minus we divide to in months 11. Victory About two minutes. Five. I would have to end my guest to end minus nine. Only up to two in minus five. No, and we have also to the n minus three, divided by two bn minus 6 22 to the n minus three minus and minus six. There are negative three post six or three sweet, too cute in the numerator. And they also have n minus three factorial divided by six factorial. You get this in minus All right, four friends. And so it's not a need. Fear how this is going. The problem is simply so. Instead of trying to some place coefficient, I'm just going to write it out. It's full form, so we have their 2nd 40 a serious solution. White two of X bread So it's easier to distinguish background two x to the are or was negative have we could take you know it's equal one. Did we have one plus son? Any unity In the end, I just want to be negative knowing negative seven Probably up to to end 11 extra again Namir we had in factorial to ease. They get three times negative to to minus four. So this is the second serious solution We know from here in the book that these two serious solutions are going to be defined now that what all positive remembers and are nearly independent on a suitable

Were given a differential equation were asked to show that the additional equation as distinct roots, which do not differ by nature and then to find two linearly independent, previous serious solutions for this differential equation. The differential equation is four x squared y doble crime plus three x y Crime plus X Y equals zero. We're really looking at this equation where X is greater than zero. Finding an initial equation off this differential equation, we'll need to write differential equating in the form of a standard differential equation with the singularity at X equals zero. So this is going to be divide through by 14 X squared y doble Prime plus x times 3/4 Why crime plus exit before why equals zero. And so we obtain the two functions p of X and Q of X were P of X is coefficient of Wipro invited by X, which is simply 3/4 in que bec since the coefficients. Why, which is excellent. And we can see that both of these he and Kyu, our analytic at X equals zero and we have the conditional equation is going to be our times ar minus one waas he is zero or plus Q of zero equals zero. Or that our times ar minus one plus 3/4 sore plus zero equals zero. So we get a defect around or R is equal to zero or negative. One plus 3/4 is negative. 1/4 r equals two on fourth. So we see that the initial acquaintance of this differential equation has distinct roots. And that's the difference between these routes is not an integer. Now we will find to linearly independent, previous Siri's solutions. So again, notice that our roots are riel and we have that are one will equal 1/4 and are to be the smaller one, which is cereal. And we have by a serum that the differential equation has a solution of reform. Why one of X equals x to the R one, which is 1/4 is the sun, a suburban xcn and equal zero to infinity, where a not is not equal to zero. And since we have that D of X and Q of X, have power Siri's expansions and all ex, it follows that this solution will be valid or X line between zero and positive. Infinity. Furthermore, because R one and r two are distinct and don't differ by an integer. There is also a second solution to this equation that is valid for zero between X between zero infinity. So you also have why to Rex equals x it er to which is gonna be zero, which is simply one times the sun from and equal zero to infinity or piece of them next to the end. This event is not equal to zero, which is valid four x in zero infinity. And we also know from this serum at these two solutions air linearly independent on zero infinity. So why one y two are linearly independent on zero to infinity? And now to find these serious solutions, I noticed that if we start with why one we have that why one prime of X is equal to if we pull the 14 inside. In fact, more generally, let's just say we had why of X equals x t r sum from n equals zero infinity Uh, a seven x to the end will be equal to the sun from n equals zero to infinity of a sedan x to the n plus r. So we have that why prime of X will be equal to some from an equal zero to infinity of and plus R times, a seven x to the N Plus or this one. And therefore we have that why double prime of X will be equal to some from n equals zero to infinity n plus R times and plus or minus one times a seven times X to the n plus all minus two And so looking at original differential equation, this tells us that four x squared times y doble prime. So we have some from n equals zero. It's an opportunity of four and plus or en plus or minus one base have been x to the n plus R minus two plus two. I just want to be in plus r plus three x y prime. So, plus the sum that equals zero to of three. And then why crime is endless or have been X The M plus or minus one plus one is X to the n plus r finally plus X y I just want to be plus the some from and equal zero to infinity. Uh, this is going to be a PSA Ben X to the n plus heart plus one equals zero. So dividing out by X to the are and then changing the third son to a son who were X to the temples are says to Damon and minus one or end. We get the some from n equals zero to infinity. Oh, four times n plus r Times n plus r minus one plus three times n plus r Ace have been X to the end, plus some from and equals one to infinity, a seven minus one x to the and minus one plus one. Rex to the end equals zero, and we see that we have now a polynomial next to the end or an infinite Siri's. So we have that the coefficients of this Siri's must all be equal to zero. We have in particular venues equals zero. We get that for n plus r times and plus R minus one plus three times and looks are equals zero and zeros. This won't be or are times are plus one. Those three are This is going to be four times R squared. So four r squared minus four plus three are This is the same as our initial equation. This is a N equals zero. This doesn't tell us anything. You that solutions, really, and he's equals one, and we have four times one plus r times one plus R minus one just simply or plus three times one plus R times base of one los A subzero as two equal zero. In fact, instead of greatness in this way, read more generally is four times M plus R man M plus or minus one most three times and plus or times a one plus a not because. Zero, because this is a n plus a n minus one for any greater than or equal one. Did you write this as a recurrence relation so that a Savannah is equal to? And then this becomes so four times impulse are times impulse are minus one plus three times acronym Pless or you get negative 8 p.m. Minus one over n plus R times and then four and plus four or minus four plus three minus one. So are substituting in r equals 1/4 since that was the greatest word for initial Beijing. And so we get that you recursive relation Eights event is equal to negative eight of the n minus seven must one over n kinds foreign plus one. So, for example, if N is equal, the one we have a subordinate point people to negative a zero over one times four times one plus one is five you have that is and is equal to then basic to is going to be negative. Ace of one over two times four times two is eight plus one is nine. So this becomes positive A not over one times two times five times nine and is equal to three. Yes, that a three is going to be negative a to over three times. Four times three is 12. This one is 13 or we have negative a not over re factorial times five times, you know, many times their team. And so in general we have that piece of Ben. It's one equal to negative one to eat and and not over in factorial times five times nine times, times four and lost one. And so we have the corresponding for Venus Siri's solution. This differential equation why one of X is equal to X to the 1/4 times. It's some from N equals zero. Well, let's just pick a not to be one and we get one plus the sound from any one to infinity of an, which is negative one again times one over In factorial I'm five times nine minds foreign plus one x to the n So this is going to be one of ours. Previous Siri's solutions find the other previous serious solution method is very similar. Except instead of plugging in our it was 1/4 here we're going to plug in r equals zero was our second group of the initial equation. So now we have that or end times and minus one plus three and a n plus and minus +10 This becomes right this out four n squared, minus foreign plus three in just going to be or in squared minus. And so that a N is going to be equal to negative a and minus one over n times and then four and minus. This is valid for n greater than or equal to one. So we have in this case instead of a seven. We're now looking for the coefficients of the second solutions. So at peace again said that minus one here and we have that is equal to one, and we have. The piece of one is equal to negative, be subzero over one times four times one minus one, which is three friends equal. To have that, he said to it's one of the equal to negative eight piece of one over two times four times to eight minus 17 which is equal to positive. Be zero over one times two times three times seven. Yeah, there's an equals three that be a history is equal to negative. Two. Divided by three times Returns to three times four is 12 minus one is 11. I just want to be negative. The zero over three factorial times three, seven tons 11 And so, in general, do this by induction. Seems pretty clear that he said Ben is going to be negative. One to the end piece of zero divided by in factorial and three times seven times times for and minus one for angry than equal one. And so, if we take the subzero to be one, we obtained these second for Venus. Serious solution. Why too equal to X to the zero simply one times one plus some from n equals one to infinity of negative ones. The end. And then you said B zero was once a times one over in sectorial times three times, seven times four and one x to the n. And so this is going to be second solution and we know by the term used to solutions of the nearly independent and are valid for all positive X.


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