Question

Urgent please show all the steps and mark all the answers and label them.. Please!!!
(1 point) The second order equation 2xy" + 5y + xy = 0 has a regular singular point at x = 0, and has a series solution 00 y= 2 Cn"+r P=0 (1) Insert the formal power series into the differential equation, we derive an equation ( -1/[(n+r)(2(r )Cox"'+ -3/2 Dejx" + Eco DC,+ 0,-1/2,0,1/40,0,-1, Cn-2)x"+r-1 = 0 =2 So we have the indicial equation =0 and a recurrence relation cn = CR-2 for n = 2, 3, ... (2) From the inditial equation we can solve the indicial roots (enter your results as a comma separated list). For any one of the indicial roots, we have cı = (3) Letr, be the smaller indicial root. Then rı and the recurrence relation becomes C = CH-2 for n = 2, 3, ... Let co = 1. From the recurrence relation, we have a solution y = x + x"+1+ X*1+2+ x +3+ x"+4+ x+5+ x"+6 + ... (4) Let r2 be the larger indicial root. Then r2 = and the recurrence relation becomes C = Ch-2 for n = 2, 3, ...

Answer 1

Qay" +54 +ay=0 nto het y = I na Co to 1 no n=0 nture I nada) 2Cp (n +31) in 131-1) + 5 Colntra N=0 n+J+1 I na no equating lowest degru term. 2 Countr-1) + 5 Co.u= 0 & 2010-1) +50=0 dar-2+5)=0 d (20+3)=0 -312 งา,ะ -312 O 1 equating Geef of a nedol op. (n+7) on +3-1) + 5 Cp (+1) + Cn-2 = 0 clnto) (2n+20 + 3) = -Cn-z On-2 n = (n+d) (an+20+3) - Cn-2 - Gn-2 39,- -312 > Ons on-tan-319) nan. 3) -Co - Co 2. I .66 = -C2 Co 2.4.5.6.9 C 4 = 4.5.2 Cg 4.5 + 2.45 2.4.5.6.9 Lt Co-J => %, -20[1- - Ona 1,-05 ana non +3) Co -co Co -Co 2.4.7. 11.6.15 CA= 8.7 2.7. 4.1

4 G a 4 21 2.4.7.)) 2,4.7.11. &.IS 님 Ad, + 18.d2 312 + > A59 JD 2. 45 3. SC-9 어 41 - + 27 2, 4.7.11 2. 4-6,7.11.18

J-) سلا ci) [2Co.) (97-1) + 5 Com] + 26).J(1+1) +5C, 69+1)%2°+ + 3 [acn{n+Dun+0-1) +SGD(147) + C7_27.20417-4 O nar- → COD (93+3) ='+ C, lau?+2J +53 +1) a'+ fan into) (an +29 43) +cra] a n=2 → Cu (974336776,19012+797 + 193% + Lehena93t2n +27143) +6n..].947 n=2 -312 J=0,