First of all, by solving this the normal way, the answer should be:
y=Ae2x+Be−x
What I have so far:
y=∑[infinity]n=0anxn
y′=∑[infinity]n=1nanxn−1
y″=∑[infinity]n=2(n−1)(n)anxn−2
Shifting the counter to make it all ∑[infinity]n=0
y′=∑[infinity]n=0(n+1)an+1xn
y″=∑[infinity]n=0(n+1)(n+2)an+2xn
Therefore, I get:
(n+1)(n+2)an+2−(n+1)an+1−2an=0
From this, my recursion formula is:
an+2=2an+(n+1)an+1(n+1)(n+2)
I then evaluated this from n=0 to n=6 but my a2 to a8 values seem to be all over the place.
a2=2a0+a12!
a3=2a0+3a13!
a4=6a0+5a14!
a5=10a0+11a15!
a6=22a0+21a16!
a7=42a0+43a17!
a8=86a0+85a18!
I researched a bit and found out that the coefficients of a1 follow the Jacobsthal sequence while the coefficients of a0 are twice the Jacobsthal sequence but I think this is wrong as it is nowhere near the power series expansion of y=Ae2x+Be−x
