5

Consider the initial value problemy" cy' + 2y = 0, y (0) = 3, y (0) = 1Find a pOwcr series solution to the above problem_...

Question

Consider the initial value problemy" cy' + 2y = 0, y (0) = 3, y (0) = 1Find a pOwcr series solution to the above problem_

Consider the initial value problem y" cy' + 2y = 0, y (0) = 3, y (0) = 1 Find a pOwcr series solution to the above problem_



Answers

Find series solutions for the initial value problems in Exercises $15-32$ .
$$
y^{\prime \prime}+y=0, \quad y^{\prime}(0)=0 \text { and } y(0)=1
$$

All right. We want to represent why as a power series. So we let it equal a zero plus a one X plus a two plus a two X squared and so on. And in general we add a sub N X to the end. Now the constant term here, a zero we know equals Y. Of zero, which we are given is equal to zero. This is one of the initial conditions we have. So we can already rewrite why as ah a one X plus a two X squared Plus A3 x cubed and so on, just like before. But we now know that A zero is 0. Okay, so from this we can find the paris series for why prime. By differentiating when we differentiate the right hand side, we get a one plus to a two X plus three A three X squared and so on. And in general we add and a sub n X to the and minus one. Now the constant term here is a one That is why prime of zero. And we are told that that is equal to one. That's the other initial condition we have. So uh we can go ahead and rewrite. We can actually rewrite Y and Y. Prime. Actually we can rewrite why as X plus A two x squared Plus A3 x cubed and so on. And we can rewrite why prime as one plus to a two X squared. No, not X squared, just X plus three A three X squared and so on. Okay. Anyway, now that we have that we can find the power series for why double prime by differentiating again and we are going to get two A two plus three times two A three X plus. The next term is going to be four times three, A four X squared and so on. And in general, ah what we are adding here is And plus two times and plus one Times a sub n plus two times X to the end. That's what we're adding in general. So we can finally find the paris series for why? Double prime minus Y. By subtracting these two Paris series for y double prime and y. So for a constant term we just get to a two for the coefficient of X. We get ah We get three times 2 A three uh minus one. That's the coefficient of X for x squared. We get four times three. A four minus a two for X squared and so on. And in general, the coefficient for X to the end, that is going to be n plus two Times N Plus one, A Sub n plus two minus a sub end X to the And All right, that's our power series for Y double prime minus Y. And we are given that we want this to be equal to zero. If we want this equal to zero, then we want the constant in every coefficient to be equal to zero. So we want to a two equal to zero three times to a three -1. equal to zero, four times three. A four minus a two Equal to zero and so on. And in general We want n plus two Times and Plus one A sub n plus two minus a sub end equal to zero. Or Taking this equation and rearranging to solve for a sub n plus two. We get that we want a sub N plus two equal to a sub n over and plus two Times and Plus one. Okay, so now we have everything, we need to find all the values of a N. We figured out earlier that we want a easier to be zero and we want a one to be one uh based on this equation, we want A 2 to be zero based on this, We want a three to be won over three times two. And okay, using this general rule from here on out, we figure out that we want a four to just be zero Because a 20, but a five A five, we want to be a three Over five times 4. So we want a five to be won over five times four times 3 times two. And then we want a six to be zero. But then we want a seven to be a five divided by seven times six. So at this point I'll go ahead and re express it is what's going on here? A seven is 1/7 factorial, A five Is 1/5 factorial. A three is 1/3 factorial. Okay, so in general in general we're getting that a sub even index two K. Is just zero. But in odd index gives us one over a factorial. Specifically the factorial of the index. So two K Plus one Factorial. Okay. Anyway, with this we can finally return finally returned to our power series for why And we find that it is equal to X plus X cube over three factorial plus X to the 5th over five factorial plus and so on. Plus X to the I suppose we can say to N plus one over two n plus one factorial and so on. Very. Anyway, this power series for Why is exactly equal to the standard power series for the function. Hyperbolic sine.

All right. We want to represent. Excuse me. We want to represent why? By a power series. So we let y equal a zero Plus A one X plus a two squared. That that that plus a sub N X. To the tucked up. Now, since A zero equals y. of zero. And we are given that Y of zero equals one. This is the initial condition that were given. We can already rewrite why? As one plus a one X plus a two X squared plus it's a paris series where the end term is a southern exterior. So if we differentiate this, then we get why prime On the left hand side on the right, we get a one plus two. A two X plus three 83 X squared. That that that the general term is N. A. Some NX to the N -1. So now based on the differential equation were given, we want to look at why plus why prime? When we add the constant terms, we get one plus a one. When we add the terms in X, we get a one plus to a two X for X squared, we get a two plus three in three. That that that so the general term here is we get a sub N plus no, sorry Not a sub n plus one. We get and plus one Times a servant plus one is in general the coefficient for X to the end. Now, since we are given that this differential equation equal zero, we want to set this entire expression equal to zero. Well if this entire expression equal zero then every coefficient has to equal zero. So We need one plus a one equals zero. We need anyone plus to a two equal to zero. And in general we need a sub N Plus and plus one a santos, one equal to zero. If we take this equation for the general coefficient and solve for a sub n pulse one, we get yes and plus one equals negative a sub and over and plus one. So following this a rule here for values of a sub N. We started a zero which equaled one and we get that A one is negative one, A two is 1/2. A three is negative 1/2 times three, then one over two times 3 times four negative one Over two times 3 times four times five and so on. This is the sequence of values of a suburban. No, okay now if this is the sequence of values of a suburban, remember ah a zero one, two, three and so on. We can see that a sub N equals negative one to the end. Over. And factorial. So finally returning to the power series we had for Why we can say that. Why equals one minus X plus 1/2 factorial times X squared minus one over three. Factorial times X cubed plus that that plus an end term of -1 to the end X to the end over and factorial. And noticed that The general term the term -1 to the end next to the end over. And factorial, this equals negative X to the end over. And factorial. So based on knowledge of standard power series, we can finally ultimately conclude that why equals mm the negative X.

All right. We want to express why as a power series? So we let it equal a zero Plus A one X plus eight to X squared and so on. And in general in general we're adding a sub n next to the. Now the constant term here, a zero. This is equal to Y. Of zero, which we are told is equal to zero. This is one of our initial conditions. So we now know that a zero equals zero. Um The power series for why prime We get by differentiating this one. This results in a one plus to a two X plus three A three X squared and so on. In general we add n a sub n X to the and minus one. The constant term here is a one Which equals y prime of zero. And our other initial condition tells us that this is equal 21 So now we know that a zero is zero and a one is one. By the way, we can also from this take get the power series for to buy prime by just multiplying every single coefficient by two. So we get to a two plus uh for a two X plus six eight. Whoops. Not 63. Just six plus six A three X squared. And so on us. Two N A n X to the N -1. All right, we also get the power series for why double prime by differentiating the paris theories for why prime? This gives us to a two plus three times two A three X plus four times three A four X squared and so on. And in general we add N Times N -1 A sub n X to the N -2 and so on. Okay, so now now that we have paris series for Y double prime to Y prime and why we can find the paris series for Y double prime minus to Y prime plus. Why? Uh Excuse me? So for the constant term we are getting two, A tu minus two A one plus a zero. That's our constant term For the coefficient of X. We are getting three times 2 A three um minus for a two plus a one as the coefficient of X for the coefficient of X squared. Um We are getting four times three. A four minus six A three plus a two as the coefficient of X squared and so on. And in general the coefficient for X to the n that we're getting is n plus two Times N Plus one. A sub n plus two minus mm Excuse me minus two times N plus one. A sub n plus one plus a sub n X. Not X seven X to the end. That's the general term of this power series. For why? Double prime minus two? Y. Prime plus. Why? Which we are given is equal to zero. Now if this power series Equals zero then we need the constant and every single coefficient to be equal to zero. So we want To a to two. A 1 plus a zero equal to zero. We won three times to a three minus for a two Plus A one equal to zero and so on. And in general we want and plus two ah Times and Plus one A sub n plus two -2 and plus one A sub n plus one plus a sub. And To be equal to zero. Now if we take this equation for the general coefficient And we re arrange to solve for a sub and plus two. What we get is that it should equal two times N plus one A sub n plus one minus ace of an over and plus two Times and Plus one. So now we can start finding values of a suburban for all in. Uh huh. We already figured out that a zero is equal to zero and a one Is equal to one. So based on this rule here we get that A two should be equal to uh one. Yeah, A three. Based on this rule here we get that A three should be equal to 1/2. And from here on out using this general rule, We get that a four mm hmm. Excuse me, A four should be equal to uh two times 3 times one half, three minus one. So it should be equal to two over four times three. All right. I paused the recording briefly to compute some more values of Osama Bin wouldn't and simplify a sub Uh a form which simplifies to 1/6. And when you compute the next few few values and simplify them, you should be getting 1/24. 1/1 21/7 20. And the pattern here that the general pattern for a suburban is that we are getting one over N minus one factorial Right to to is two factorial six is three factorial, 24 is four factorial five factorial six factorial. This is the value of a suburban. So now we can finally return to our power series for why. And we get that why? Based on these values equals X plus x squared plus x cubed over two factorial plus X to the 4th over three factorial and so on. And in general we're getting X to the N plus one over And factorial based on this power series we have here and knowledge of standard power series, we can conclude that Y equals X times E to the X.

Doing great. You're going to solve a problem. Number four. Here we assume the resolution off the phone y equals Sigma Chemical 20 to infinity. See and exist to it, and we differentiated legal that void icicles sigma a naked one to infinity and CNN Express toe end minus one. So we need to solve the differential equation. Takes minus three violation plus two white equal to zero, substituting that we will get like X minus three in do sigma canonical toe one to infinity N. C and access to an minus one plus two sigma and equal to 0 2050. Seeing access to an equal zero sigma a negative one to infinity N. C and excess two and minus three Sigma a nickel toe. Want to infinity M C and access to end minus one plus two Sigma n equals zero to infinity. See and access to an equal to zero minus three Sigma Eneko toe 1 2050 n. C. And access to and minus one plus Sigma n equal to zero teen 50 and plus to see and expressed when equal to zero. Uh, and so like minus three sigma a negative zero to infinity and plus one seeing plus one expressed when Plus Sigma and they go to zero to infinity and plus two seeing access to an equal to zero. So the it will come like Sigma and I go to zero to infinity and plus two seeing minus three, 10 plus one see and plus one access to n equals zero for the song Toby zero, We Need Life and plus two scenes. Three into endless one c n plus one equals zero c n plus one necklace and plus two divided by three and do n plus one dot c so they can't right like See, you know will be unknown. So see one equal to zero plus two. Developed by three into zero plus one into sinner, which is by three C. Note. C two equals one plus two. Developed by three into one less one see one, which is equal. Took one by three c. Note. C three because two plus two divided by three into one plus one C two, which is equal to for by 27 c not then c four equals three plus two. Limited by three into three plus one see took, which is equal to 55 81 c. No foreseeing it was and plus one divided by three rest to N. C. Not so Why equals Sigma and equal to zero to infinity. Seeing access to it, which is ableto see not Sigma n equal to zero to infinity and plus one they were by three days to him into access to it. It is equal to see you, not Sigma and equal to zero p. Infinity X by three holder student Let's say not sigma any Go to zero to infinity and x by three Older parent, which comes to be like C zero Sigma and Nico 20 to infinity X by three For the parents. Yes, C zero Sigma n equals zero p infinity. It's de by be It's off. It's by three. Hold up our end so we will get like C zero indoor one by one minus x by three plus zero x The by the X off one developed by one minus x by three from for which comes to be like please see, not developed by three minus six plus C not x D by B X, off three by three Minus X, which is equal to three into three minus six. See? Not divided by three minus X The whole square was trees, you know, that's developed by three minus six. The whole square. Logistical tau nine C not developed by X minus Trade a whole square, which is see by X minus strata, whole square. So why equals C developed by X minus? Traitor! Who's got? That's the no for question. Thank you.


Similar Solved Questions

5 answers
4) Find the free, forced , steady and transient parts of (show all the steps)Xtx+3Ox 20 X(O)= 5 X() =/0
4) Find the free, forced , steady and transient parts of (show all the steps) Xtx+3Ox 20 X(O)= 5 X() =/0...
5 answers
Identify and sketch the graph of each surface:6x4y + 3-225x2 9y2522225flxy) = y + 2f(x,y)1+2x7 +2y
Identify and sketch the graph of each surface: 6x 4y + 3- 225x2 9y 2522 225 flxy) = y + 2 f(x,y) 1+2x7 +2y...
4 answers
At relativistic speeds_ two events that are simultaneous in one frame of reference will be:A) simultaneous in all frames of referenceB) simultaneous in another frame that is moving in the opposite directionsimultaneous in the same frame of reference
At relativistic speeds_ two events that are simultaneous in one frame of reference will be: A) simultaneous in all frames of reference B) simultaneous in another frame that is moving in the opposite direction simultaneous in the same frame of reference...
5 answers
For the equation of the parabola given in the form (y-k)2= 4p (x - h), (a) Identify the vertex, value of p, focus, and focal diameter of the parabola. (b) Identify the endpoints of the latus rectum (c) Graph the parabola (d) Write equations for the directrix and axis of symmetry: Express numbers in exact, simplest form_(y+4)2 =- 16k-4)
For the equation of the parabola given in the form (y-k)2= 4p (x - h), (a) Identify the vertex, value of p, focus, and focal diameter of the parabola. (b) Identify the endpoints of the latus rectum (c) Graph the parabola (d) Write equations for the directrix and axis of symmetry: Express numbers in ...
1 answers
Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$\log _{5} 0.65$$
Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$\log _{5} 0.65$$...
4 answers
If an equation is true for all values of its variable, it is called an _______________.
If an equation is true for all values of its variable, it is called an _______________....
5 answers
Score: 0 of 1 pt40i8complete}3.4.35-TQueshon HelpThe data Ue riqht canain tte sale aqaretleIor 11 Z0 raqionscounine0.55 0.98 0,30 2 24 4,35 0,62 270 1.78 1,53 0,87Conptele patts and (b)Compuie the Ducxil:LAI Incan and nonulalin wandard devltnnTOWIA craile AarGrIne populalan medn (Round Iwa ducIn-I Licusnocduo
Score: 0 of 1 pt 40i8 complete} 3.4.35-T Queshon Help The data Ue riqht canain tte sale aqaretle Ior 11 Z0 raqions counine 0.55 0.98 0,30 2 24 4,35 0,62 270 1.78 1,53 0,87 Conptele patts and (b) Compuie the Ducxil:LAI Incan and nonulalin wandard devltnn TOWIA craile AarGr Ine populalan medn (Round I...
1 answers
Solve each formula for the indicated letter. $A=\pi r^{2},$ for $\pi$ (Area of a circle with radius $r)$ (FIGURE CANNOT COPY)
Solve each formula for the indicated letter. $A=\pi r^{2},$ for $\pi$ (Area of a circle with radius $r)$ (FIGURE CANNOT COPY)...
5 answers
Polarity and solubilty predictions forBenadryl/Diphenhydramine?1. Draw your molecule2. For each central atom in your molecule, decide if polaror nonpolar.3. For polar central atoms , color the central atom red if itforms hydrogen bonds and green if it forms dipole-dipoleinteractions.4. Make a reasonable prediction will be soluble or notsoluble in water and give reasoning.
Polarity and solubilty predictions for Benadryl/Diphenhydramine? 1. Draw your molecule 2. For each central atom in your molecule, decide if polar or nonpolar. 3. For polar central atoms , color the central atom red if it forms hydrogen bonds and green if it forms dipole-dipole interactions. 4. Ma...
5 answers
SubmitBequest AnswcPart €Find the magnitude of the magnetic field this electon produces at the point C .AzdSubmiRequeat AnawePart D Complete provious part(s)
Submit Bequest Answc Part € Find the magnitude of the magnetic field this electon produces at the point C . Azd Submi Requeat Anawe Part D Complete provious part(s)...
5 answers
Let 𝑆 = {-1, 0, 1, 2} and 𝑓(𝑥) = ⌊ 𝑥2 3 ⌋. Find 𝑓(𝑆)
Let 𝑆 = {-1, 0, 1, 2} and 𝑓(𝑥) = ⌊ 𝑥2 3 ⌋. Find 𝑓(𝑆)...
5 answers
The pole supports a 22-Ib traffic light: Using vectors Determine the moment of the weight of the traffic light about the base at A_I8 0
The pole supports a 22-Ib traffic light: Using vectors Determine the moment of the weight of the traffic light about the base at A_ I8 0...
5 answers
Sx3 If F(x) = Indt, what is F' (x)?Provide your answer below:F' (x) ~
Sx3 If F(x) = In dt, what is F' (x)? Provide your answer below: F' (x) ~...
5 answers
The unit tangent vector of the curve r(t) = 2sin(2t)i + 2cos(2t)j tk isSelect one:vcos(2t)i Vi5 sin(2t)j _ Kskcos(2t)isin (2t)j _ XkC.None of these answersD_4cos(2t)i + 4sin(2t)j _ k4cos(2t)i 4sin(2t)j = k
The unit tangent vector of the curve r(t) = 2sin(2t)i + 2cos(2t)j tk is Select one: vcos(2t)i Vi5 sin(2t)j _ Ksk cos(2t)i sin (2t)j _ Xk C.None of these answers D_ 4cos(2t)i + 4sin(2t)j _ k 4cos(2t)i 4sin(2t)j = k...
3 answers
(35 points) Let $ {p(r) € Ps | p(1) = 0 and p(1) = 0}. (p (z) is the derivative of p(z) with respect to r.) $ is subspace of P; (you do not need to show this) _ Find basis for S_
(35 points) Let $ {p(r) € Ps | p(1) = 0 and p(1) = 0}. (p (z) is the derivative of p(z) with respect to r.) $ is subspace of P; (you do not need to show this) _ Find basis for S_...
5 answers
(a) Use differentiation to find power series representation forfx) = (7 + x}Rx) = > D E0What is the radius of convergence, R? R =(b) Use part (a) to find power series forf(x)ftx) = _What is the radius Of convergence, R =(c) Use part (b) to find power series iorWhat iS the radlus of convergence;Enhanced Feedback
(a) Use differentiation to find power series representation for fx) = (7 + x} Rx) = > D E0 What is the radius of convergence, R? R = (b) Use part (a) to find power series for f(x) ftx) = _ What is the radius Of convergence, R = (c) Use part (b) to find power series ior What iS the radlus of conve...
5 answers
QUestouWhich statement Is corrcct about ANAEROBIC cellular resplratlon? Chooxd cocicetHunun mwsckes perform inietobik celluin respiration:Anterobic cel ular respintion tne sane tntrfemenaboninorganic molecule used (he final electon scccplar.Oxdstivr phosphonxation occursAn organic mo"cule used Js Ul: final elecbron scceptorPrcvious
qUestou Which statement Is corrcct about ANAEROBIC cellular resplratlon? Chooxd cocicet Hunun mwsckes perform inietobik celluin respiration: Anterobic cel ular respintion tne sane tntr femenabon inorganic molecule used (he final electon scccplar. Oxdstivr phosphonxation occurs An organic mo"cu...

-- 0.020617--