5

1) Given the equationy" 3ry' + 6y = 0, use the power series method to find two its fundamental Find the solutions near I = 0: recurrence formula for an; F...

Question

1) Given the equationy" 3ry' + 6y = 0, use the power series method to find two its fundamental Find the solutions near I = 0: recurrence formula for an; Find formulas for the two power series solutions; Find the raflius of convergence of both series solutions.

1) Given the equation y" 3ry' + 6y = 0, use the power series method to find two its fundamental Find the solutions near I = 0: recurrence formula for an; Find formulas for the two power series solutions; Find the raflius of convergence of both series solutions.



Answers

Determine two linearly independent power series solutions to the given differential equation centered at $x=0 .$ Also determine the radius of convergence of the series solutions. $$y^{\prime \prime}-2 x y^{\prime}-2 y=0.$$

But it's a little too. Why crime us wise. He could stop. Okay, we know that white crime that's equal to the sum from energy quits one infinity of end times a and extra and minus one plus. Why, which is the sum from and as you could to girl to infinity of a end Sex extra pair of endless is equal to that. Okay, now the indexing. So this is the one we get. Just hold onto the same plus some from energy equal to one infinity of a N minus one extra and a minus one is equal to job. So this becomes a summation from N is equal to one infinity. It's factor out an expert opinion on this one. Times to end and plus hey, And like it's one we got to go. So we get that two and a n plus a n minus one is you don't r a n is equal to a n. What? It's one over two. Okay, so it's all for a one that's equal to one over to negative a knot. What is the two that's equal to negative one over 281 What a one waas a good one of virtues that we get negative one over to practice too. A knot a tree is a sequel to the one or 2 to 3 A. Not so you could see a powder from this. It seems like with this squaring or we're taking the power to of negative point over too. So our general form is why is equal to a not summation from n is equal to zero infinity one over for actually have one over to power and extra her oven.

Okay. The problem they gave us here is just gonna be wide, Double prime plus two x squared Y prime plus two x y equals zero came a p of x and Q of extra. Both polynomial Z and I are a radius of convergence with a spear. Infinity. Um, next I would just We're just going to plug it in like normal. So, uh, for my wide double prime, I'm gonna set my K. But, uh, my end is equal to K plus two. And so I should be able to write. That is K equals zero to infinity. Okay, plus two times K plus one times Eisuke plus two times X to the K. So the whole point is to get that value extra. Okay, that would have plus two. And because we're multiplying by X squared here, we're gonna have ah x to the n plus one. So we need to use an equals K minus one in this case. Okay, so that's gonna leave us with K equals two strengthening OK, minus one, sometimes a C K minus, one times X to the K, and similarly for the my wife turn already is K minus one for it as well. So we're gonna started. K equals one. Go to infinity. We have a set of K minus one. Don't we have to hear and then extrude the K No. Zero Now, in order to combine like terms here, we're gonna have to go ahead and evaluate cable zero and one for her first term. And que goes one for a last term case of booking in zero. I'm gonna have ah, uh, to a set to plugging in. K is equal to one. I'm gonna have six a some three x plus two a not X. It's a non X comes from my third expression. Sorry, Those are both Times X. They were gonna plus summation starting from K equals two. Since we've already evaluated, K is one in zero to infinity and they just put all of your terms here except for your ex to the K that has to equal zero, you know, for each of my, um, coefficients to be zero Oh, that means my face up to past equal zero and that my ace of three has to equal negative 1/3 a no no and then the bracket term. We can solve that for zero and it comes out to be a sieve cave close to equals. Negative two K number K plus two times K plus one. I'll leave that algebra to you times a so K minus one. Because we have a difference of three from our K plus two to K minus one, we're gonna have three linearly independent systems. Maybe that independent, but at least two it's the three n plus one. And if we have a three n plus two But I three n plus two because it just deals with, um, starting with a sub to an ace of five. And it is expression involving a sum tube than these are all just gonna be zero. Okay. And then plugging in, I'm gonna get ah, ace of three in fairly complicated expression. But go ahead. And I tried simplifying it a little bit. I'm sure there's more ways to simplify. So get negative two to the end times for turn seven times. All right, you go until you get to three and minus two. They all over two times. Three times. Five times six. I'm starting. Um, you gotta go to three in minus one times three in can. All of that is times a subzero. Okay, Ace and three n plus one. It comes out to be very similar. It's gonna be negative to to the eighth Power. And my name Raider will go starting from three. Ah, about terms. Seven times posted a three, and then five times, eight times, 11 times That. That, uh Uh huh. A three in minus one. That's because of the three to the numerator just to kind of help me with my denominator. That's just gonna be three times for I just wanted that three in the numerator, uh, times six times. Seven times that. That three. And times three and plus one. That's all the army times A to the first. Okay, so using those, um, my three n we'll use that large part of my expression, right? Why one of X? It's just equal to a not terms of summation of an equal zero to infinity of the large brackets that go with three. And and of the X to the three end. Okay, then why two of X ray do the same thing was gonna pull the expression in front of the A one that's gonna be equal to a one. Terms of information from an equal zero to infinity of that expression that matches three n plus one one of the X to the three n plus one. Okay, that's it. Thank you very much.

Okay, So, uh, in this equation have wide, double prime minus y equals zero. Uh, with the equations in this little chapter that would give us that Pierre vax is equal to zero, since there is no why prime term and cue of X is equal to and I get one. So because these are both, I guess he could They're both polynomial on that, then. All of the points for X is equal to X, Not will be ordinary points. Um, And then their power. Syria's expansions about X equals zero are gonna be valid for all X. Um, and said the power serious solutions will converge for all real eggs. So there is no our value in this case because every point would converge. Okay, so it takes care of the radius of convergence part. Okay, So from from here, I'm just gonna let why Ivax equal Teoh information from an equal zero to infinity of a sub in times X to the end. And that's gonna give us that. Why? Double private bags that would equal to summation from in equals two to infinity of end times and minus one times a seven times X to the n minus two. Okay, so, uh, plugging this back into my original equation, I'm just gonna have this summation. And for my first equation, I'm gonna set my K value equal to K plus two. And some have k equals zero to infinity. My n is equal to K plus to someone. Okay? Plus two times K plus one times a sub k plus two. And then I'll have X to the K, which is the whole point is trying to get this extra the K term they have minus, I mean for this summation. I'm just gonna set my an equal to case okay equals zero to infinity and have a so okay next to the king. Okay. And so because that equals zero, that means ah, the summation of all of our terms in front of the extra decay, but also have to equal zero. Um, so we could right. Okay. Plus two times K plus one ace of K plus two must equal. It's OK. Okay. Anyway, so therefore we can write Eisuke. Plus to you is equal to one over. Okay. Close to times K plus one times a. So Okay, it's no. Had to look at the two cases, Uh, for the even case, um, A to vocalist. Even it is gonna be equal. Teoh, 1/2 times a night. A four a sub for Is it gonna equal 1/4 tons? Three times a two, which is gonna be 1/4 factorial. Hey, Sim Zero. Okay, we can keep following this pattern so that a sub two in because these air even is gonna be, ah, 1/2 n factorial times a sub zero. Okay, for the odd case, we're gonna have a three. Yes, equal to 1/3 times to times a one. Ace of five would equal whenever, five times for times a sub three, uh, which is just equal to 1/5 factorial and then a said one. And so we can write a sub two and plus Warren, because we're dealing with odd numbers, you're just gonna be equal to one over two n plus one factorial times, eight of the first. Okay, so therefore, we can write these as two separate linear equations. So for my even, I'll have line one of acts is gonna be the equal to the summation. Uh, well, we can can pull of the a not term some estimation from in equal zero to infinity 1/2 times and factorial times x to the to end Because we're only talking about the even powers and for on expression we can, right? This is why some two of X equals a one just following the same kind of pattern from an equal zero to infinity, the one over two and plus one factorial times x to the two n plus one S O R Y setbacks gonna be the equal to the sum of these two linearly independent power. Siri's Sorry, I've last of Mexico's. Why one of axe? That's why two. Okay. And there you have it. I want to my one and we have there, son. Thank you very much.

Okay, so this problem gives us the equation. Why? Double prime a minus X squared y crime minus two x y equals zero. Both p of x Anke of X are polynomial is and some of our value it's just going to be equal to infinity because it will converge for all next. Okay, so in this problem, um, we can write our initial y of axe as the summation from in equal zero to Infinity of Ace of Ben, thanks to the end. So taking all the way down the second derivative on, then put him back in. We're gonna have the summation from an equals two to infinity of end times and minus one times a seven times x to the N minus two. A round of minus. We're gonna make multiplying by X squared n equals one to infinity of in times a seven. It's Ah X minus one x to the n minus one. We're gonna have extra the end plus one in their last time, we're gonna be multiplying by X against we have another x to the n plus one. But where money's too the estimation from in equal zero to infinity of a seven times x to the n plus one. You know, that's gonna equal zero. Now, I don't want things all to be just X to the case. So for my 1st 1 I'm just steps into K plus to infer end. And the 2nd 1 is gonna be okay. Minus one. Pretend in the 3rd 1 is well, K minus one for in. It's a signal that and I'm gonna have the summation from an equal zero to infinity. Okay? Plus two times K plus one times a set K plus two times X to the K. There were no minus summation from in. I believe it's gonna be in equals. Oh, sorry. That's not going to be in since we K's on a big major change. Hopefully that mess you up, uh, we're gonna have K So have K minus one equals one. So we're gonna go from K equals two to infinity, OK? Minus one times aces K minus one times X to the King. I never would have minus two. So, in summation, we're going to say this is not gonna be K equals one to infinity of a c k minus one x to the K equals zero. It's only issue now is we got to get our summations to all, um, be the same And someone changed them all The b k equals zero. In order to do that, um, I have toe subtract out the term That would be the zero in the one term. So when K is equal to zero for my middle term, let's say, ah, we would write our middle term. I'm gonna put parentheses around, exclude that negative there as the summation from K equals zero to infinity. So K minus one. Yes, OK, minus one toe extra. The k when they went to subtract out, uh, when we plugged in one. So they plug in one, we get zero memento, worry about that. But when we plug in zero, but we're gonna get negative line. So I have to add one times a city we're putting in zero. Okay, so it gives us the negative one term somebody back this up and I'm just gonna treat everything to start at K equals two. So I'm not gonna mess with this on any longer. I'm only gonna mess with my first and last ones. This is from the 1st 1 I'm a zero term is gonna be two times one. He said to ace it too. Uh, Times X to the zero. And they were gonna have Plus, Now we're gonna plug in one so have three times to which is six, sometimes a sub three times X. And they were not plus our summation. Now, from K equals two, our second one's already done sort of last time. We just have to plug in for our first term. So plugging in our first term, we're gonna have minus two were playing in one. It's just gonna be a subzero at times X and then we can say minus two times of summation from K equals two to infinity. Okay, so that's a mess. But like we got for one Keiko. Zero cables one and then from two on day. So from K equals for K equals zero. Um, we're just gonna have to a to o n. K. Equals one. We're gonna have six a three minutes to a not Times X, and then K greater than or equal to two. You have our summation. So we're gonna have plus the summation from K equals two to Infinity of bracket. Alright, K plus two times K plus one. Ah, yes. Okay. Plus two, we have minus K minus one times a set of K minus one. The then minus two case of K minus one bracket X to the K equals zero. Okay, so from that we know that if all of this is gonna equal zero, then when K is equal to zero, we just have and to a to Since we need all of the coefficients to visi room when K is equal to zero, that means to a two is gonna equal to zero. So a sub two is gonna be equal to zero for K equals one. We we know that six a three minus two a knot has to equal zero on. And so we know that six a three. It's gonna equal 1/3 a zero. Okay. And then for this, the larger term we can write a relationship of a sub K plus two. This is for K greater than or equal to two is equal to. So we added the ace of K minus line, and we're gonna have if we pulled it out, we're gonna have K minus one plus two so would be left with K plus one and divided by cave plus two times K my plus one K plus ones cancels. We're just left with one over K plus two times a so K minus one. Okay, so we're gonna try to get some linearly independent. Um, the is, But it looks like because Eisuke plus two is directly related to a C K minus one. That means we're gonna do factors of three. So it's not just gonna be even an odd it's gonna be three and three n plus one and three in plus two. Okay, Say only wants to. So I have to investigate. Kind of See what's going on. I'm thinking, because this a two is equal to zero. Uh, that this pregnant. Cancel out one of these. Okay, so three. And we're talking about a subzero. Um, so that's the next term. Up would be a sub three. Okay, so we have a some three we already know is equal to 1/3 a subzero. It's a case of six. It's gonna be equal to 1/6, uh, times. Hey, sit three. It's a nine is 1/9 times 1/6 times, 1/3 times a subzero. So looks like we're gonna Backpackers and three summits. Asep three. And it's gonna be won over three times. Six times, dr 60.3 in times a subzero. In fact, we could probably even write this as Oh, I mean, my pauses and see Yes. So we can write this as one over three to the end and then times in factorial a subzero. But it now three n plus one. That means we'll start a sub one. So the next term would be a sieve four. That s a four is just gonna be equal to 1/4 terms. Ace of one. The next term up would be a Step seven. Just give me 1/7 times for terms. A sibling swing. Pride. Right? This one is a sub three n plus one. I don't know if there's a shortcut for this one. It's just gonna be won over four times. Seven times dot dot three n plus one case of one. Okay. And in three n plus two, I'll start with Ace up 2 to 0. That ace of five is one of her five times zero. Yeah. So this one is completely gonna cancel out to zero so we can write. This is two terms. So the 1st 1 Why did one of X This is gonna be our our three end value, The ace of zero. Ah, from an equal zero to infinity of one over three to the end times in factorial times x to the three n became wide two of axe There's gonna be equal to ace of one times a summation from an equal zero to infinity of one over four times. Seven times that, I thought three n plus two times all the way and stop at three n plus one. Ah, and then that's gonna be times X to the three n plus one. Okay? And then does your to linearly independent power Siri's? Ah, to this differential equation. Thank you very much.


Similar Solved Questions

5 answers
When muscles attach to bones, they usually do so by a series of tendons, as shown in the figure (Figure 1). In the figure, five tendons attach to the bone. The uppermost tendon pulls at 20.0? from the axis of the bone, and each tendon is directed 10.08 from the one next to it:Figure1 of 110? 2 102 4 108 1 1084209TendonsBone
When muscles attach to bones, they usually do so by a series of tendons, as shown in the figure (Figure 1). In the figure, five tendons attach to the bone. The uppermost tendon pulls at 20.0? from the axis of the bone, and each tendon is directed 10.08 from the one next to it: Figure 1 of 1 10? 2 10...
5 answers
If the molar mass of NOxl-is 46.01 g/mole, then the value of Xis
If the molar mass of NOxl-is 46.01 g/mole, then the value of Xis...
5 answers
1. The average bond enthalpies for 0-0 and O-0 are 146and 496 kJ mol-1 respectively. Whatis the enthalpy change in kJ for the reaction below? H-0-O-H (g) H-O-Hg) + YO-Olg)
1. The average bond enthalpies for 0-0 and O-0 are 146and 496 kJ mol-1 respectively. Whatis the enthalpy change in kJ for the reaction below? H-0-O-H (g) H-O-Hg) + YO-Olg)...
5 answers
ChEK CEAGuON"Soling for 0 reactontusima cecmicalequalenMamy e ttilizur $MadaFeacnna phosphorlc OIC (H,FOJmmoniaAnmoniun phosptale ((NH,),"04) (NH,}'imconart aiedrentsmmor unt phospratproducedReD of 2 7prosphonc acid?LtalMnseSure Yjur Onswer hascorrect numta4n Iicant digits
ChEK CEAGuON" Soling for 0 reactontusima cecmicalequalen Mamy e ttilizur $ Mada Feacnna phosphorlc OIC (H,FO Jmmonia Anmoniun phosptale ((NH,),"04) (NH,}' imconart aiedrent smmor unt phosprat produced ReD of 2 7 prosphonc acid? LtalMnse Sure Yjur Onswer has correct numta 4n Iicant dig...
5 answers
L-Y{c-16+2s} =? (Hint: use partial fraction technique to simplify the given function)Select one: 3et_ 3e-2t 3et-Ze-2 %et-Ze 3et- 3e-2
L-Y{c-16+2s} =? (Hint: use partial fraction technique to simplify the given function) Select one: 3et_ 3e-2t 3et-Ze-2 %et-Ze 3et- 3e-2...
5 answers
Pnce InbounMuttCtdnnmdEEp Tel 1crlronoaot ! Folno -UedtonedureKoer]
Pnce Inboun MuttCtdnnmd EEp Tel 1 crlronoaot ! Folno -Uedtonedure Koer]...
5 answers
Question 2: Prohlem solvingProfits ((S*100) Sea- ~tood caterNumber of cratesSea-food caterySea-food eateryConstruct the tables for 7l = 3: " = and n = Show the calculations. (11.5 Marks?Stage n =3 (2 Marks) f8 (ss)Stage(7 Marks)fz(s2,*2) Pz(x2) f362f (s2)StageMarks)J1(s1,*1) = P1(*1) + Ji(3 -_fi (s1)Based on lhe lables you consbucled wrle down lhe oplimial allocalion_ and Ihe maximium lalal prolil: (3 Marks)
Question 2: Prohlem solving Profits ((S*100) Sea- ~tood cater Number of crates Sea-food catery Sea-food eatery Construct the tables for 7l = 3: " = and n = Show the calculations. (11.5 Marks? Stage n =3 (2 Marks) f8 (ss) Stage (7 Marks) fz(s2,*2) Pz(x2) f362 f (s2) Stage Marks) J1(s1,*1) = P1(*...
5 answers
Use polar coordinates to evaluate the area of the region bounded above by the unit circle _ +y? = land below the line y 2
Use polar coordinates to evaluate the area of the region bounded above by the unit circle _ +y? = land below the line y 2...
1 answers
Let $\mathbf{u}=\langle 1,2\rangle, \mathbf{v}=\langle 4,-2\rangle,$ and $\mathbf{w}=\langle 6,0\rangle .$ Find (a) $\mathbf{u} \cdot(7 \mathbf{v}+\mathbf{w})$ (b) $\|(\mathbf{u} \cdot \mathbf{w}) \mathbf{w}\|$ (c) $\|\mathbf{u}\|(\mathbf{v} \cdot \mathbf{w})$ (d) $(\|\mathbf{u}\| \mathbf{v}) \cdot \mathbf{w}$
Let $\mathbf{u}=\langle 1,2\rangle, \mathbf{v}=\langle 4,-2\rangle,$ and $\mathbf{w}=\langle 6,0\rangle .$ Find (a) $\mathbf{u} \cdot(7 \mathbf{v}+\mathbf{w})$ (b) $\|(\mathbf{u} \cdot \mathbf{w}) \mathbf{w}\|$ (c) $\|\mathbf{u}\|(\mathbf{v} \cdot \mathbf{w})$ (d) $(\|\mathbf{u}\| \mathbf{v}) \cdot ...
5 answers
La EaplanLelunnuechmgMennatMa hilnmAFR c4151 EJAto ma IRA4m 78 _7J dopeelaeu ue uennal MJuoHot rererol the IRA wi} centain { IbA nol etndun Fnen mmtenetreJulai depog (Typa = Eno e nunean|pand 1
La Eaplan Lelunnuechmg Mennat Ma hilnmAFR c4151 EJAto ma IRA4m 78 _7J dopeelaeu ue uennal MJuo Hot rererol the IRA wi} centain { IbA nol etndun Fnen mmte netre Julai depog (Typa = Eno e nunean| pand 1...
5 answers
Refer to rectangles. Complete the tables. $p$ is the perimeter.(TABLE CANNOT COPY)
Refer to rectangles. Complete the tables. $p$ is the perimeter. (TABLE CANNOT COPY)...
1 answers
COORDINATE GEOMETRY Given each set of vertices, determine whether $\square E F G H$ is a rhombus, a rectangle, or a square. List all that apply. Explain your reasoning. $$E(-7,3), F(-2,3), G(1,7), H(-4,7)$$
COORDINATE GEOMETRY Given each set of vertices, determine whether $\square E F G H$ is a rhombus, a rectangle, or a square. List all that apply. Explain your reasoning. $$E(-7,3), F(-2,3), G(1,7), H(-4,7)$$...
5 answers
HRSE is a kite. What are the coordinates of point $R ?$CAN'T COPY THE FIGURE
HRSE is a kite. What are the coordinates of point $R ?$ CAN'T COPY THE FIGURE...
4 answers
A buffer solution is made by mixing 250.0mL of 0.15 M HCHO2 with100.0 ml of 0.20 M LiCHO2. The Ka for HCHO2 is 1.8 x 10^-4.Calculate the pH of the solution afterthe addition of 100mL of 0.3M HCl to the buffer.(Show work and write your answer in 3 sig fig)
a buffer solution is made by mixing 250.0mL of 0.15 M HCHO2 with 100.0 ml of 0.20 M LiCHO2. The Ka for HCHO2 is 1.8 x 10^-4. Calculate the pH of the solution after the addition of 100mL of 0.3M HCl to the buffer. (Show work and write your answer in 3 sig fig)...
5 answers
Determine if each function is even, Odd or neither. Justify your answer.J= 2x' _ 3r+5f() = 314+5
Determine if each function is even, Odd or neither. Justify your answer. J= 2x' _ 3r+5 f() = 314+5...
4 answers
(2 pts) Let A be an Xn R 2 2 matrix such that AT = -A_ For what values of n is det (A) 07 Justify your answer.
(2 pts) Let A be an Xn R 2 2 matrix such that AT = -A_ For what values of n is det (A) 07 Justify your answer....
5 answers
1(1-) conetnc [IEA inequality 0-1 1Fu] ntenalom{
1 (1-) conetnc [IEA inequality 0-1 1 Fu] ntenalom {...
5 answers
Let F-< 4r8y8 22 32*yP 22 2c*y3 2>If Fis conservative find fsuch that Vf = F and use this to find Jc F - dr where C is the straight line segment from (2,1, 1) to (1,1,-1)
Let F-< 4r8y8 22 32*yP 22 2c*y3 2 > If F is conservative find fsuch that Vf = F and use this to find Jc F - dr where C is the straight line segment from (2,1, 1) to (1,1,-1)...
5 answers
Crper Lr jearay [Da (rth 04 cnaanutrrom#4Bint sotg Ksou9 Bo4Dh Eroim Bau D B
Crper Lr jearay [Da (rth 04 cnaanutrrom#4 Bint sotg K sou9 Bo4 Dh Eroim Bau D B...

-- 0.022336--