Question

Question 5 (60 points) For the following first order ODE: 1) x'y' + (3x - 2y = 0, Xo = 0 DO 3X2 @O 2) (1-x)y + 2y = 0, xo = 0 3) (x^ - 6x + 10)y +(12 - 4x2y = 0, Xo = 3 a) What are the points of singularity for each specific problem? We undefined b) Does this ODE hold a power series solution at the specific xO? Justify your answer, if your answer yes, then do as follows: i) Compute the radius of analyticity R around each Xo and the intervallo which the power series solution is valid; ii) find the corresponding values of the coefficients in power series
please solve with power series. part b for 2and 3 is what im really struggling with

Answer 1

- for differial equation y' + b(x) y = 0 0 A point, a, is point of singularity if p(x) is NOT continuous at ä. ats - o aso q as, ag, ... an are n singularities of 0 then radius of analyticity R = Minį l ao - Q₂l, 106-Agl, ... 16-anly power series and interval on which is valid is (-R, R) (2-) (1-x) y' + 2y = 0,2 = 0 | 8) 8 tỷ 8 - 0 : blæ & (CS scanned with X = 1 is point of singularity. CamScanner

boint solution - 16.) : x=0 is NOT the singular - ODE has a power series - at No =0. (i) azt is the only singular i R = Mir ļ 1%-al} - Min į 10-11} = 1 point . in L radius of analyticity and Interval I on which the is valid in I=(-4,1) power series E = (ii) Let the power series solution about to do; y = Cn (-x)" In oc" (as a6=0 here) n = 0 os y = non enim n=1 Scanned with CamScanner

- but this in given ope. c" ² Cn och =0 = 0 xn- en i nGXn-1 + 2 (2-x) n0 n=1 .00, +2san x-0 >> Žnen an! - na an-1+1 n=0 n=1 & nenoch-t - na och ta 8 cn och=0 n=o N=L nal Now, as į na och = 0+ § na och nal n=1 = 0 co x + Engan n=1 n0 : @ can be written as, ngoch- s noon +2 E Cn och = 0 nal n=0 n=0 Scanned with CamScanner

nxn- n=1 - no - 3 6 xeno - (n -2) (n 2" 30 - In est sum of 3 replace n by nth E Here nit to ao l earlies) n+:1 to ao Cafter replacement) > nilod to ao = n: o to ao E: 3 becomes, - ž (n+1) Cnt X - E (n-2) en oc" =0 nu-1 - " ne 0 no n = 0 Ca(net) Grey 2" - Z. (n=1) (&* =0 - a len +) Com - (n=) (n}" =0 ;n=0,1,2,.. (n+1) Cats - (n-2) (n = 0 1 Cnt - (n-1) # (n+1 This is recurrence relation n=0 Ę = -20o = -20o Scanned with CamScanner

nad 6 = ģu - * (-2)6 – Co n=2 Cz - og lg = 0 nas CH = C2 = 0 (as (p=0) in Cq=0=4=G=-.. Now as y= 2 e conoch isy - Có +612 + ( x + C923+.+--- :y-co + (-a)Cox + Co xo +0.23+0.3++ .. : - co (2-24+) 7-6(1-28+&Tis the power series solution of (1-0) y' + 2y = 0 about xo=0. Scanned with CamScanner