Question

13. Consider the differential Equations y" + xy + 3x²y =0. a.) Use the power series expansion about Xo=0, y = { anx", to find the recursive formula. MO b) Find the first 4 terms of the general solution. You do not need to seperate y, in terms of ao and Yz in terms of ai

Answer 1

Qi Consider differential Equation y" + xey' + 322y = 0 about 2=oy- and to a) 970 Use power Series expansion Find recursive formula SOW Lety-Ź anan 7120 Let y y'a n-1 En an x na as n=2 get .nl en-2 + n20 nej En(n-1) an zen-2 putting these values in giron ditterential Eq", we È n(n-1) an a x ² nana + Bxc2 anx"=0 Ž n(n-1) anze nanx + 3² anx (Replace n by 072) Replace -2) xn È nanan 3 Žan-2 Š 22+2)(n+1) Antz? 12=2 n+2 n n-2 + nzo 11) in by noz 7-2 n 22 t n=2 n=1 n=0 n=2 202 + 6932 + É 6+2)(n+1) Anta? an taxt { nanan n=2 + 8 & ameto -> 20+ (683 +0,7x + Ž (0+2)(n+1) antet n an +3an-2 n=2 [R.H.S is x are o Zero Cofficient af all power of =0 202 69z+a, = 93 - = 0 - Qy 6 CS Scanned with CamScanner

anta az = 0 A N 6 11 Sol y nzo 5 anan Frost Four Terms of and (+2) +i) Antz + n an +32n-2 = 0 =2,3, -(nant 3 an-2) n = 2,3,- (22+1 6+2) 23 -Qi 6 Thus represents the recursive formula the Fisst 4 terms of the general sole Generd sol y = alot ax + az x² + ₂ x3 + First Four terms of sole y= ao + ax + aa x² + az x3 ta y = lot aux 0.x? 6 y = as tan from qu = -(242 +390) (Q+1 2+2) ay = - 300 -ao y = x4 4 9427 as 23 xt ook + aq x4 a x3 + auxy 6 n=2 2 as = = 0 4 3 а, у Ao t aux ao x 6 CS Scanned with CamScanner