- #1

- 28

- 4

- Homework Statement
- Solve the DE in terms of a power series

- Relevant Equations
- power series

I believe I am doing everything right up until the point where I have to try and find a recurrence relation. I honestly have no idea what to do from there. I've listed my work in getting the powers of n and the indicies to all match. Any help appreciated.

Here is the original DE:

##y^{'}-(x-1)y=0##

##\sum_{n=1}^\infty na_n x^{n-1}-x\sum_{n=0}^\infty a_n x^{n}+\sum_{n=0}^\infty a_n x^{n}=0##

##\sum_{n=1}^\infty na_n x^{n-1}-\sum_{n=0}^\infty a_n x^{n+1}+\sum_{n=0}^\infty a_n x^{n}=0##

##\sum_{n=0}^\infty {(n+1)}a_{n+1} x^{n}-\sum_{n=1}^\infty a_{n-1} x^{n}+\sum_{n=0}^\infty a_n x^{n}=0##

##2a_0+\sum_{n=1}^\infty {(n+1)}a_{n+1} x^{n}-\sum_{n=1}^\infty a_{n-1} x^{n}+\sum_{n=1}^\infty a_n x^{n}=0##

##2a_0+\sum_{n=1}^\infty \left[{(n+1)}a_{n+1} -a_{n-1}+a_n\right]x^n=0##

##a_0=0##

##a_1=-2a_2##

##a_2=-a_3##

##a_3=-2a_4##

##a_4=-\frac 5 3 a_5##

##a_5=-\frac 9 4 a_6##

And this is where I get stuck.

Here is the original DE:

##y^{'}-(x-1)y=0##

##\sum_{n=1}^\infty na_n x^{n-1}-x\sum_{n=0}^\infty a_n x^{n}+\sum_{n=0}^\infty a_n x^{n}=0##

##\sum_{n=1}^\infty na_n x^{n-1}-\sum_{n=0}^\infty a_n x^{n+1}+\sum_{n=0}^\infty a_n x^{n}=0##

##\sum_{n=0}^\infty {(n+1)}a_{n+1} x^{n}-\sum_{n=1}^\infty a_{n-1} x^{n}+\sum_{n=0}^\infty a_n x^{n}=0##

##2a_0+\sum_{n=1}^\infty {(n+1)}a_{n+1} x^{n}-\sum_{n=1}^\infty a_{n-1} x^{n}+\sum_{n=1}^\infty a_n x^{n}=0##

##2a_0+\sum_{n=1}^\infty \left[{(n+1)}a_{n+1} -a_{n-1}+a_n\right]x^n=0##

##a_0=0##

##a_1=-2a_2##

##a_2=-a_3##

##a_3=-2a_4##

##a_4=-\frac 5 3 a_5##

##a_5=-\frac 9 4 a_6##

And this is where I get stuck.