Question

Seek power series solution of the given differential equation about the given point x0; find the recurrence relation.
(1-x)y'' + y = 0; x0 = 0

Answer 1

I'll give it a try.

(1-x)y"+y=0 at x0=0
A. Check for singular points.
Write as y"+(1/1-x)y=0
and find Lim x->0 (1/1-x) = 1
Therefore x0=0 is not a singular point.
B. Assume y=∑an x^n for n=0 to ∞
then
(1-x)y"=∑n=0 to ∞ (n)(n-1)an x^(n-2)
=∑n=2 to ∞ (n)(n-1)an x^(n-2) -∑n=1 to ∞ (n)(n-1)an x^(n-1)
Now switch indices
=∑n=0 to ∞ (n+2)(n+1)an+2 x^(n) -∑n=0 to ∞ (n+1)(n)an+1 x^(n)
Add to y gives
y(x)
∑n=0 to ∞ (n+2)(n+1)an+2 x^(n) -∑n=0 to ∞ (n+1)(n)an+1 x^(n)+∑ n=0 to ∞ an x^n =0

This gives
an+1 = [(n)(n+1)an+1 - an]/[(n+2)(n+1)]

The coefficients an are dependent on values of a0 and a1, which are arbitrary constants depending on the given initial conditions.

Check my arithmetic.