Question

(1 point) In this problem you will solve the differential equation (+7)y"+11xy' - y=0. x" for the differential equation will converge at least on the interval (-inf.-sqrt(7)] (1) Ey analyzing the singular paints of the differential equation, we know that a series solution of the form y = . (2) Substituting y = . *" into (x2+7y" + 11xy - y = 0, you get that Multiplying the coefficients in x through the sums E Reindex the sums Finally combine the sumns -- • * . . |--- The subscripts on the e's should be increasing and numbers or in terms of n. (3) In this step we will use the equation above to solve for some of the terms in the series and find the recurrence relation. (a) From the constant term in the series above, we know that (b) From the coefficient of x in the series above, we know that (c) From the series atave, we find that the recurrence relation is - C o to a (4) The general solution to (x + 7)y"+11xy-y=0 converges at least on and is y = co( + + + + ... + ( x+ + + x +...) (5) Solve the initial value problem (x+7)}" + 11xy' - y=0 WO) = -6 70) = 8

Answer 1

R-PIFE (y t-2n y'- n=2 う Pat abone Values C0 Sn(nプ)enxと 十11x5nenx -2. Cxta) - D - ax= N3 n= ハ=D MullApiyiry the sum cofficent in x Bmayh he + 72nin-Denx nen-cn= n-a nin)en ニ0 ハ= Reindux he sum 00 n=l N-0 00 au

00 00 メt 2ain-) Cnx' +4 Cg +432メ+2 (nt>) LnナDnt n-2 の=2 - o - Co-CIX Gx duelly Combuning ハ-Q From conslaut eun n eveh Know dhal FRes Co we knon from Cerffievent f xn eiuer ab 109 LE4.243 &euer abone, we fnd that ewusieut J tn-リム 40nt) nta)chtr Cn -nti) Mt) Cnt n 7

10 Saso 70 8.2, 520 tne STution s Co t Cix+ Gxxlu Cot Gx tloyx-Co3304x C 82,82 2 176 82320