Question

diff eq

the problem states that to solve the given linear initial-value problem use the power series method.

please include intermediate steps

(x² - 1)y"+ 3xy + xy = 0, y(0) =4 , y'(0) = 6

Answer 1

, y lol = 4 - 7' 10) = 6 Q solve Jup by hower series method [22-1) y" + 3x y' + xy =0 - 0 som Tom comparing std. diff egn P/X2 y" + Q(x) y' + R(x) y = 0 - P(x) = a* –, 0(t) = 3% where: Ria) = 2 so let P(x =0) = -1 O x=0 is elringe odinary point series soim of 0 is y = Ž anamo diff w. rotor z' = ¿ am n x" n -1 8พ มะ B aguin diff wire to a ě n/m-1) an xm-2 y" = พะ = mimai) am **** ma 2

given that Y(0) = 4 .. a lot a,(0) + a20 = yo) 4 = a + => 4 = ao y' lol = 6 2 9'(%) : 8 - | a, + Q. 3, (0) + 3, 0° 6 = a, there fine coon {a, ao from o (x²-1) " + 3xy! + xy =0 C using @ , & => (x²-1 ) 8 mln-il an xn-2 + 3x Š namxml ma) n2 => n(n-1) am 2" - , Vana) cm 29. 2 + ž zmam 2" + evento com 2m+=0 n=2 mai

= (m - 1) - 2 => S - 3 n + 1)(x + 2) a + 2 1 nel na + 230 am 2" + * = 0 => (m) am X" -[204 +6052 + en av mine smrti") +[30,2 + B 39 an ] + [dox + , 0m**) =0 → - 242 + x1-603 +39, +00) te [mIm-1) an – (**)) (m+2) antz+ 30 Cm + Cm-,7 x? =0 m2 In 2 on a2=0 comparing -2a2 = 0 => -603 +39, +90=0 - Az = + 134,+90) - (346+u) a = "

& mem-v) am 1 nln-il an ante = -lna)/n+2) On+ 2 + 3 anant any = 0 1 l mln-il an + 3 nam tam., 1 4) (+2) > chillen we in pnu 23 an + amar ] an-, 7 m), in an + (n )/n + 2) -> am +2 = 1 + but n=2 an = o + + x6 =/ 12 but n =3 as = 3 az + hors an = 3x "68 +0="/ so required sum is glal = a + a,x+ a₂x²+ az x² + aux" tanas tano **** ts3 hp + y2 € + x H + x9 + 1 = (x)