Question

methool: series uSe 2+ aroundd XG 5 X 2.

Answer 1

Answer:

The given differential equation is

5 2.x 1 y y 0 h-

The above equation can be rewritten as,

......... equation (1)

2 5ry( 2.x -1)y = 0

As per power series formula of differential equation,

y'(x) na"-1 n (n 1) d-2 y" (x) y (x) n n-0 n-1 n-2

Putting the above values in the given differential equation (1) we get,

nana" (2x 1)an 1) an25a n-1 n (n n-2 n 1 n-0

х п (п - 1) а,а" 2а,т"+1 — У ал"%3D 0 5па," + + n-2 п-1 n-0 п-0 M

We need to change the third series to make coefficient of x as n

х п (п - 1) а,а" 2ал-12" - У а,." 3D0 5nа," + + n-2 п-1 n=1 п-0 M

We can start the series from n=0 as initial terms in first three terms are zero,

After more simplification we get,

n(n 1a 5nan 2a-1 1] 0 0

n2 4n)an 2-1 1]0 0

(n24n)an2an-1 10 where n 1,2,3...

1-2an-1 here n= 1,2, 3... an n(n4)

for n = 1,

$a_1 = \frac{1}{5}-\frac{2}{5}a_0$

a2 _ 541 25 25 +

$a_3 = \frac{1}{5}-\frac{2}{5}a_2 = \frac{1}{5}-\frac{1}{25}+\frac{4}{125}-\frac{8}{125}a_0$

Therefore,

$a_k = \sum_{1}^{k-1}\frac{1}{5^k}(-1)^{k-1} \ \ + \frac{(-1)^{k-1}2^{k-1}}{5^k}+\frac{(-1)^k2^k}{5^k}$

ΣΣ- (1)-12k-1 1 (-1)- y(x) (1)2k 5 k-1 aρ 5k Α 5k 0 Τρ