question #2 only

ANSWER
Given differential equation is,
Here, we can take
L(y) = x
2
y
’’
+ (¼ + x
2
) y = 0
For the above given equation the singular point will be 0. So it is a form of Bessel’s equation of order n.
Since x=0 is a singular point of the given equation so we can assume that the solution of the above differential
equation is of the form,
y(x) = ∑
So, its first and second differential will be,
y
’
(x) = ∑

y
’’
(x) = ∑


On replacing the obtained values in the given...