Question

First determine a recurrence formula for the coefficients in the (Frobenius) series expansion of the solution about x = 0. Use this recurrence formula to determine if there exists a solution to the differential equation that is decreasing for x > 0. *?y'' - x(7+xy' + 16y=0 What is the recurrence relation for a ?

Answer 1

NOLO x2 gios Differential equation x@y" – X (776)+l6y=o -O y" - (1+x) + 16 2y =0 Now comparing caith standard second ordeo D:£ Y" +PK) y' + Q(x) y = 0, we have PIX) = -(14%) and Q(x) = hem -> at porót x=0, both pix) and Qw) does not exist => x=0 is zo singules point . Now (01.31 deo (2-0) P(x) = -1772) and (2-0)?Q(X) = 16 both XPW) and x20\x) is exist at x = 0 => x=0 is reguleo Singules point Then Solution exiót about x=0 o Then cele can apply foobenius method to solup eqo let yex)= xo tanen = 2 an xnto - is Solution of Era On, men salisfies equo, com haur no ทะ) no I Go (DE) n+8 )x 1 +2 no -TI no (n + +162 no ze Ian Cov) (nr41) 287-2. (1270) { ga(147) X" 7010. E color) (nt 1) **- 13 any an empty xong emotion Lan ((n +9) 2 (0+Y) – 469+) +16] 2140 E an (n7)x++++! Za [(14228 16+9)+6] 20+? - Ian (110) ***+20 Eign E () -472 mi** Zan ento) xhi7720 no no ned

Now 9 ceaill find indicial equation . To find indicial equation equating th lowest dogore Coefficert. we have from ean ③ 90(8-4)=0 9ot0 =(1-4)2=0 r=4,4, positive and repeated root for Recarrenco.gelatico, equating to zero the coefficent of ze het in eq" , cees have an K(+) - 4]2 - 090-1 (n+81) = 0 an = (1+-1) [(n +)-472 7 comento

Now put r=4, in ean ③ we haux an = (n +4-1) an-l (n+4-4)2 Tan= 9+3 ani (n+44 Anez. put 1=2, 12 = , ] n-3, 1 = = a + 1 = 0 n=4, 194 = 1 2 2 3 = 4.5 12.22.3.42° Similary we weill find a5, 96,97 - - - from ean ☺ put r=4 , come haup 418)= 24 [ 20 + Q1X +9gx? + 93+3+99xf+--1 4.5.6.1.40 n-4, yo = re* [&o + 0x + .20 x ? + VIII, 6, 13+ -- gen) = 939 [1 + 4x + 4; 5x2 + 4.5.4** --] =a.14 - -- TX + 4.5 2 12.22 X2 + 1.5.6 12 12.2? 32 This is Required Boletion of ODE by using Foobenous Method