Question

4) xy" + y' – xy = 0,x, = 0 a) What are the points of singularity for each specific problem? b) Does this ODE hold a general power series solution at the specific xO? Justify your answer, if your answer is yes, the proceed as follows: Compute the radius of analyticity and report the corresponding interval, and Identify the recursion formula for the power series coefficient around xo, and write the corresponding solution with at least four non-zero terms explicitly written If your answer is no, then please elaborate why there is not a general power series solution, and proceed to solve it using the Frobenius method

Answer 1

4. Given that ry + y ² ay-0,260... a. Discussion about singularity at 4050 : Re-write the given equation. c y" - y = -0. comparing in with y + praly' + G (n)y=0. ..pin) = 1 G11=-1. Here (x-0) pim)-1 and (n-Oja(m)=- Showing that (2-0) pin) and (9-054) are analytic at a=o. iaco is a regular singular point. b. let y = 8 cm etho where co to be solution y = 8 Con Cr+m) x ktm- y e am clcum) " (Ktm-1) m=0 maOD m=0 ktm- m =0 the equation becomes 2 Cm(Kim) (km-1) akm-2 a Con Citmix & me ktm = 0. & Cm(kam) (K+m-1) + 8 cm (1 mly 4th mo Ktm- mo Υη mar mco 2 cm ((Atm (Kem) + (km) } & Kam - 8 cm altmth o. MED

& Cm (Kim Romno -I am a m m=0 o rita which is an identity equating to aen the coefficient of the Smallest power of namely & K, (2) gives the indical equation Corao or co. (Coto. o. SO (20,0 equal roots. eguating to Zero the coefficient of the next smallest power ott. ie af (2) gives G6+1)=0 G=0. For recurrence relation, equating to zero the Coefficient of a ktm t es (2) gives - cm (kam) cm-250- or cm = Com -> irim) Putting m=23,4,. . - e-Com t =0, Cu = Cety)ÝCikte)} ( 2) is m=0 and so on. We putting these values in y= 8 cma mtic y = x [co + G x + b 2 + (3 X 4...] ya [coterapeut o Y = Continent a ter ]

C1020 Putting K o and replacing to by a "in cu) gives i yra lite --J-au- a second solution is a where y is given by ©. Ditterenthting par Kally wirto k3@gives 24 - Cox e loga [ 14 2 2 + c trudny + Co ak [0 - 2 ta borrow (122 7 4 ) Putting k o , and replacing to by b in 6 gives Core-o = bloga[it amet - au 2) (K+2jYB4YjV = b[ uloge & ( 2 ay ( 1 ) - 1] - bv -> @ FO the required solution is y=autby anonin where a and b are arbitrary Constants by using Frobenius method