I need to find a series solution to the following simple differential equation x 2

Tristian Velazquez

Tristian Velazquez

Answered question

2022-06-21

I need to find a series solution to the following simple differential equation
x 2 y = y
Assuming the solution to be of the form y = a n x n and equating the coefficients on both the sides, all the coefficients turn out to be zero which is definitely wrong.
Any help is appreciated.

Answer & Explanation

EreneDreaceaw

EreneDreaceaw

Beginner2022-06-22Added 20 answers

if you let y = n = 0 a n x n
x 2 y = n = 0 a n n x n + 1 = n = 0 a n x n
so equating coefficients on each side gives you a 0 = 0 and a relation
a n + 1 = n a n
which does indeed lead to all coefficients being zero
What if you supposed that y = a n x n
then a 0 could equal A say and then you would get 1 a 1 = A
You should then be able to recover the answer that Autolatry found using separation of variables

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