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We shall look for a power series solution around x o = 2;These issues are settled by the theory of power series and analytic functions 12 Power series and analytic functions A power series about a point x0 is an expression of the form X n=0 ∞ a n (x − x0) n = a 0 a1 (x − x0) a2 (x − x0) 2 (24) Following our previous discussion, we want to know whether this infinite sum indeedSolution 1 First assume y 2 =ux 2, and y 2 '=2uxu'x 2 and y 2 ''=u''x 2 4u'x2u Substituting into x 2 y''2xy'6y=0 we get x 2 (u''x 2 4u'x2u)2x(2uxu'x 2)6y=0 and simplifying by adding like terms we get u''x 4 6u'x 3 =0 We reduce the order by w=u' to get w'x 4 6wx 3 =0 Now dividing by wx 4 and rearranging, we get

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(1-x^2)y''-2xy' 6y=0 power series solution

(1-x^2)y''-2xy' 6y=0 power series solution-C) express the solution satisfying y(0) = 1, y′(0) = −1 in terms of y1(b) 2xy′′ (3−x)y′ − y = 0 Solution (a) Write the differential equation in standard form y′′ − 1 2x y′ 1 4x y = 0 We have P(x) = − 1 2x and Q(x) = 1 4x, so x = 0 is a singular point Taking limits we get lim x→0 xP(x) = −1/2 and lim x→0 x2Q(x) = 0 Therefore x = 0

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 By taking the derivative term by term, y'=sum_{n=1}^infty nc_nx^{n1} Now, let us look at the differential equation y'=xy by substituting the above power series in the equation, Rightarrow sum_{n=1}^infty nc_nx^{n1}=x cdot sum_{n=0}^\inftyc_nx^n by pulling the first term from the summation on the left, Rightarrow c_1sum_{n=2}^inftync_nx^{n1Find second solution to {eq}x^22xy"6y=0 {/eq} given that {eq}y1=x^2 {/eq} Solution of differential equation The given differential equation is a second order linear differential equationIe, a solution of the form (258) y(x) = X1 n=0 a n (x 2) n Our task is then is to determine the coe cients a n so that this y(x) indeed satis es (257) Now already the rst two coe cients are determined by the initial conditions To see this note (259) y(x) = P 1 n=0 a n (x 2) n = a

Answer to Solve 2xy''(x1)y'2y=0 by method of power series Get more out of your subscription* Access to over 60 million coursespecific study resourcesAnswer (1 of 4) Step 1 The trick is to make the change of (independent) variable t = ln x, to reduce the equation to a linear differential equation with constant coefficients, which presumably you already know how to solve Then x = e^t and dt/dx = 1/x If you are careful with the product ruleSolve The Differential Equation 1 X Y 2xy 6y 0 By Using The Series Solution Method Homeworklib Y 22y12xy=0, from which y 2 2 (x1)y1=0, from which y=1x±√ ( (x1) 21) by the quadratic formula or alternatively, y=1x±√ (x 22x) Either way, you can pick any value of one variable that makes sense in the expression, to get the corresponding value

 Let y be a polynomial solution of the differential equation ( 1 − x 2) y ″ − 2 x y ′ 6 y = 0 If y ( 1) = 2, then the value of the integral ∫ − 1 1 y 2 d x is First of all I can solve it by conventional method, which is long and provided this is a competitive exam question, the maximum time I can give is 2 − 3 minsMy Patreon page https//wwwpatreoncom/PolarPiFull Power Series method for solving Diff Eqs Playlist https//wwwyoutubecom/watch?v=L93PLTrMGYI&list=PLsTB 0 (x)y" b 1 (x)y' b 2 (x)y = 0 Then the equation has a general solution either of the form or of the form where c 1 and c 2 are roots of the indicial equation and A and B are arbitrary constants

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6C6 Find two independent power series solutions n anx to (1 − x 2 )y − 2xy 6y = 0 Determine their radius of convergence R To what extent is R predictable from the original ODE?Answer (1 of 5) The equation \displaystyle{ (1x^2)y'' 2xy' 2y = 0 }\qquad(1) Since we have no obvious way to find any particular solution of (1) so we should try to find its general solution in the form of a power series as follows \displaystyle{ y = C_0 C_1x C_2x^2 \dots C_nx^2Can anyone give me an advice that helps me to solve this kind of DE $$ x^2 \cdot y'' 2x \cdot y' 2y = 0 $$ knowing that $$ y_1=Ax{B \over x^2} $$ is a solution

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If possible finod 1 and 12) 2 Find a power series expansion about 0 for a general solution to the given differential equation (z 1)y', _ y 0, solve the differential equation (1 – x?)y" 2xy'6y=0 by using the series solution method solve the differential equation (1 – x?)y" 2xy'6y=0 by using the series solution methodMath 115 HW #11 Solutions 1 Show that the power series solution of the differential equation y0 −py = 0 is equivalent to the solution found using some other technique The basic idea to finding a series solution to a differential equation is to assume that we can write the solution as a power series in the form, y(x) = ∞ ∑ n=0an(x−x0)n (2) (2) y ( x) = ∑ n = 0 ∞ a n ( x − x 0) n and then try to determine what the an a n 's need to be

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Get an answer for 'Find a general solution using power swries ((x^2)1)y'' 6xy' 6y =0' and find homework help for other Math questions at eNotesAnswer (1 of 2) x1/y =2 2xy3y=2 x=21/y = (2y1)/y 2(2y1)/y 3y = 2 4y2 — 3y^2 =2y QUADRATIC EQUATION 3y^2 6y 2 =0 3y^2 6y 2=0 a=3 b=6 c=2 y= 6{eq}(1 x^2)y'' 2xy' 2y=0, {/eq} into a Maclaurin series to get a recurrence relation for the coefficients of the solution series By selecting appropriately the first two coefficients we'll

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Set m= n2 n2, then rewrite the first sum in terms of m, then replace m with n 1 level 2 SterlinMerlin Op 7y So then in the following summations i need to put them in terms of m as well?Given the equation $$ 2 x y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} = 2 x^{3}$$ This differential equation has the form y' P(x)y = Q(x)(a) 4xy′′ 2y′ y = 0;

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8113 Find two linearly independent power series solutions to the dif ferential equation y′′ 9y = 0, and determine the radius of convergence for each series Also, identify the general solution in terms of familiar elementary functionsHow to Solve a Differential Equation with Series (x 1)y'' xy' y = 0 with y(0) = 2, y'(0) = 6If you enjoyed this video please consider liking, sharing,Experts are tested by Chegg as specialists in their subject area We review their content and use your feedback to keep the quality high 100% (1 rating)

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X 1 y′ 2 x 1 y = 0 Then P(x) = −3x x 1 is singular at x = −1 and Q(x) = 2 x1 is also singular at x = −1 If R is the radius of convergence of a series solution, then R > x0 −(−1) = 2 (b) Write the equation in standard form y′′ − 3 x2 1 y = 0 Then P(x) = 0 is not singular anywhere but Q(x) = −3 x2 1 is singular at x = ±i See the answer See the answer done loading (1x^2)y''2xy'2y=0 Find the power series solution in powers of x show the details Expert Answer Who are the experts? Stack Exchange network consists of 178 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers Visit Stack Exchange

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Answer to Use power series to solve the initialvalue problem y''2xy'4y=0, y(0)=0, y'(0)=1 By signing up, you'll get thousands of stepbystep See below Assuming a power series solution like this y = a_0 a_1 x a_2 x^2 a_3 x^3 ldots = sum_0^oo a_n x^n implies y^' = sum_1^oo n a_n x^(n1) qquad qquad y^('')= sum_2^oo n (n1)a_n x^(n2) With this power series y''2xy'y = 0 implies underbrace(sum_2^oo n (n1)a_n x^(n2))_(=sum_0^oo (m2) (m1)a_(m2) x^(m)) 2x sum_1^oo n a_n x^(n1) sum_0^oo a_n x^n= 0 implies sum_0^oo (n2) (n1)a_(n2) x^n 2 sum_1^oo n a_n x^n sum_0^oo a_n x^n= 0 implies 2 a_2 sum_1^oo (n2) (n1Soo in the second summation for eaxaple I would have Sum from m=2 to inf of (mr2) * (mr3) * a_ (m2) * x m2r 1 Continue this thread

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Find the first five terms of the power series solution for xy′′ yln(1 − x)=0 (24) and determine a lower bound for its radius of convergence Solution First we determine the radius of convergence Write the equation in standard form y 2 x2 y=0 (39) Thus p= 2 x 1 2 x,6C7 If the recurrence relation for the an has three terms instead of just two, it isExtended Keyboard Examples Upload Random Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music

17 Differential Equations Page 335 336 6 X6 B Cn 2 2n 1 N 2 N 1 Cn 17 5 Power Series Solutions 17 31 Exercises 17 5 In Exercises 118 Use Power Course Hero

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Exercise 215 For the equation \(x^2 y'' x y' = 0\text{,}\) find two solutions, show that they are linearly independent and find the general solution Hint Try \(y = x^r\text{}\) Equations of the form \(a x^2 y'' b x y' c y = 0\) are called Euler's equations or Cauchy–Euler equations They are solved by trying \(y=x^r\) and solvingODEs Find the first four terms of the power series solution to the IVP y"2y'y=x, y(0)=0, y'(0)=1 To check our answer, we find the solution using th2xy9x^2 (2yx^21) (dy)/ (dx)=0, y (0)=3 \square!

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9 Find general solution of the following di erential equations given a known solution y 1 (i) (T) x(1 0x)y00 2(1 2x)y 2y= 0 y 1 = 1=x (ii) (1 2x)y00 2xy0 2y= 0 y 1 = x Solution (i) Here y 1 = 1=x Substitute y = u(x)=xto get (1 x)u00 2u0= 0 Thus, u0= 1=(1 x)2 and u= 1=(1 x) Hence, y 2 = 1=(x(1 x)) and the general solution is y= a=x b=(x The solution is y=c_0 cosxc_1 sinx Let us look at some details Let y=sum_{n=0}^inftyc_nx^n, where c_n is to be determined By taking derivatives, y'=sum_{n=1}^inftync_nx^{n1} Rightarrow y''=sum_{n=2}^inftyn(n1)c_nx^{n2} We can rewrite y''y=0 as sum_{n=2}^inftyn(n1)c_nx^{n2}sum_{n=0}^inftyc_nx^n=0 by shifting the indices of the first summation by 2, Rightarrow sum_{n=0}^infty(n2)(n1 #(k3)(k2)a_(k3)(k1)a_k=0# for #k = 0,1,2,3, cdots# This infinite recurrence equation needs the a priori definition of #a_0, a_1# which can

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Get stepbystep solutions from expert tutors as fast as 1530 minutes Your first 5 questions are on us!Nxn−1 6C2 Find two independent power series solutions P a nxn to y′′ −4y= 0, by obtaining a recursion formula for the a n 6C3 For the ODE y′′ 2xy′ 2y= 0, a) find two independent series solutions y1 and y2;5 Find a power series solution about x = 0 of the differential equation y′′ y = 0 6 Find a power series solution about x = 0 of the differential equation y′′−4y = 0 7 Find the recurrence relation for the terms in the power series solution about x = 0 of the differential equation y′′ xy = 0 8

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