Question

Number 11 please. And please explain the final step to your y= equation

10–14 SERIES SOLUTIONS Find a power series solution in powers of x. Show the details. 10. y" - y' + xy = 0 11. y" - y' + x’y = 0 12. (1 - x?)y" - 2xy' + 2y = 0 13. y" + (1 + x2)y = 0 14. y" - 4xy' + (4x2 – 2y = 0 ons

Answer 1

. ). 9" у х* y = 0 #uu 9- 4ot ax +4 +ux +4° +а хаах +4, + + - .--.- у" - 24,+ (0,1 + 12 * + 20 - 300 x + --- C-9 = -а, - 29 х - 30х-част е 50s - ху - оо х* +а -а, ---

- 9" - y +x o (242-4 ) + (саз-ге) х -бо, - зе, х* + (204, +94 + а) х* + (3sag tsag + 1 Equating the coeltidents on Its & RHS - 29, -а, -о- 4 = 2а, -а, = a 64. – 2а, во – 20 = 6аз - = а, -а, 14 - 20 + 0 0 2 12а - 302 –ао а - аз ао 12 4 - 4 - ) )

Now 20g - uanta, -o *200g = naural as= u( al ao sola 20 309-5a5 ta,=0 300g = sag az 3046 = 15 (saya) - al Therefore substituting az, az, ay, as, a in terms of ap, a, in ep of y

+-- У= a +4 + 4 4 4 + + + 92 % +4х+аг х2 + а, « , -а) * 17an ач ( Ans