Question

2. Use the power series method to solve the following initial-value problem: y" + 2xy' + 8y = 0 with y(0) = 3 and y(0) = 0.

Answer 1

The given initial-value problem is Y"+ 2xy +8y=0 with y(0)=3, 0) = 0 let the power series solution y= ² anh Then = { nan x - * * and y" n(n+1), xena Substituting for y, y, y" in the given equation we get E non=3a, zin2 22. Ši nan 21 +05 and to no 21 : 1 1) and + ananan & sanx = 0 1-an ana (2n+8) as an o n-2 anx 1 [2 (0-2) + 8 Janez 2-2 =0 n-2 -o a replacing on by n-2 n -2 n2 (-)an x2 E n=2 (2n+4) On-2 2002 = 0 Že nen-nan ada 2 Ž (204) an. x2 =o Jan 2-2 2 (na)ant (2n+ 4 an 2 2 - 0

componing both sides we got (n-Don & (2014) ango Anz2 (non n.2 Now putting n=2,3,4,5,6...... 2,-- au= - 12 A2 = (-1). Ewao ana - 14 a3 = - 16 - 17 an 12 =400 age16 Q4 .600 aqua 05 -- penting all these values in y = arpen we got y = a + a, x + az x2 + 02 23 & au&ors & +4 = 0 + 0,X *(-40) x2+(- 0,1)33 +8=003176 49.28 44,31-34. 4o1x2 + 3x VI. 12 ha + putting y()=3 in 0 we have putting y'(o)=0 in 4 we have a 199=0] ['co al xo] Now put a = 3, 0,50 in 15