Question

Use power series methods to solve the initial-value problem

y''-2xy'+8y=0

y(0)=3 y'(0)=0

You must show your work and the power series method You only need to show the first four nonzero terms of each series in your answer

Answer 1

n-2 R(n-Dann Z (1+2) (n+Dan+ 2² y_2xy+8y=0;4(0=3; YO=0 Let: yaxy Emotys so, ylzný +8y=0 In Ch-Danu 2 I znanx +I 8 and = 0 ao=3 ay=0 En(n-Dahan-2 an Inanan +8 I amun = 0 n=0 uzo n m. h=0 n-a nze no n=0 nzo ELEM +) (0+2) cm+28+ 1h+D (2+2) an+ 20 + (2m+8) ann E2m+8) am 2=0 So, the coeffient of ren is equal to zero, 30 (n+) (M+2) an+2+ (-2m+8) an=0 if n=0; 20+8000 » az=-400=-12 if n=1) fazt 69 ħ az=-a=0 it u=2; 12ayt ya=0> Zay=-02= 12 it n=3; au=4 2005+29z as=as = 0 The solution is an y = ao tape + azut tahtaunut = 3-12-22 7 um " + sor