Calc 3 / sort of differential equations question
Solve using a power series:
I am given y′′ − (x^2 + 8)y′ + y = 0 centered around x0= 3
I've taken the derivatives and plugged them into the original ODE. Where I'm stuck is simplifying the second term (which is more of a Calc 3 question I think): -[(x2 + 8)] multiplied by the sum from zero to infinity of na(sub n) (x-3)^n-1

Question

Calc 3 / sort of differential equations question
Solve using a power series:
I am given y′′ − (x^2 + 8)y′ + y = 0 centered around x0= 3
I've taken the derivatives and plugged them into the original ODE. Where I'm stuck is simplifying the second term (which is more of a Calc 3 question I think): -[(x2 + 8)] multiplied by the sum from zero to infinity of na(sub n) (x-3)^n-1

astrophysics · Answer

Ah interesting so yes this is a differential equation, and you've never solved using series correct?

astrophysics · Answer

It's been a while for myself as well, but generally we have $$y(x) = \sum_{n=0}^{\infty} c_n(x-x_0)^n$$ if it was centred around 0 would be so much simpler hehe, I think it'll be ok. Then $$y'(x) = \sum_{n=0}^{\infty} n c_n (x-x_0)^{n-1}$$ and $$y''(x) = \sum_{n=0}^{\infty} c_n n(n-1)(x-x_0)^{n-2}$$

astrophysics · Answer

And then you should be able to set it up pretty nicely, make sure your index is correct ;)