Question

☺ (x²+2)y" +3 xy'-y = 0 Sol. -te altex *-* = ! ----**-**+1 = "h

Answer 1

n(n-1) ausen Here the given Differential equation is G?+2) 7" + Bryl- y = 0 is an clearly x=0 ordi nary point o has solution of the form Let us suppose y = { anach nro @ Y' - E nanana nco L ound n-2 Z y = 14 n(n-1) and no in 7, 8, 9 substi get tituting (x²+2) I non-1) ana m2 + 3nt nanah Sansen no no h + 3nann no nro nro n-o E -{ansenzo {{nen-1) + 3n-13 ank" + 2 n(n-1) anzania sa-2)(n-3+ 31h-)=120.2304+ 25 en 1) 2,24 o no no nra CS Scanned with CamScanner

n-2 n-2 E m2- 62-51 +6+31 -6- 1) Qu-22 + {2n(n-1) and T n=2 n=2 n-2 n=2 Zero [362_20 -1 ) am2 + 20 (n_nan Jan Eacrating the coefficient of 2n-2 to (M?- 20 -1) an-2 + 2060-1) an zo we have n²an- ana amaz n=2,3... 2n(n-1) 4-u-lao : (2-1) 2.2 4 9-6-lai 아 G Az = 2.3(3-1) 7 16-8-as 7. 24 ao AL au . 96 2.4 25-10-1 aza 2.5.14 7 20 aza 7 al 120 1 a5 CS Scanned with CamScanner

. The power series solution is given by yeni = E anun nco - aoff a x tan2 faz x² + auk" tasus = aut alin + taon? - an - Thanh + 72 anfore 17 18 x 4 - - -/+ a, (x-t x² + 2 x = 20 (1 + x2 4 *-) - coy, tay2 U2 wture У 2 4 Love" + x + 3 Love x 3 у 45. They yny aoy, tay 72 and cu two arbitrary constant ao are where CS Scanned with CamScanner