Why do we use different methods for finding power series solutions?

[matematikuvol]

Why sometimes we search solution of power series in the way:
$$y(x)=\sum^{\infty}_{n=0}a_nx^n$$
and sometimes
$$y(x)=\sum^{\infty}_{n=0}a_nx^{n+1}$$?

 

[tiny-tim]

hi matematikuvol!

Why sometimes we search solution of power series in the way:
$$y(x)=\sum^{\infty}_{n=0}a_nx^n$$
and sometimes
$$y(x)=\sum^{\infty}_{n=0}a_nx^{n+1}$$?

no particular reason …

sometimes one gives neater equations than the other …

they'll both work (provided, of course, that y(0) = 0)

 

[matematikuvol]

I think that in the case when
$$\alpha(x)y''(x)+\beta(x)y'(x)+\gamma(x)y(x)=0$$
if $\alpha(0)=0$ you must work with $\sum^{\infty}_{n=0}a_nx^{n+k}$, but I'm not sure.

 

[tiny-tim]

I think that in the case when
$$\alpha(x)y''(x)+\beta(x)y'(x)+\gamma(x)y(x)=0$$
if $\alpha(0)=0$ you must work with $\sum^{\infty}_{n=0}a_nx^{n+k}$, but I'm not sure.

but that's the same as $\sum^{\infty}_{n=0}b_nx^n$ with $b_n = a_{n-k}$ for n ≥ k, and $b_n = 0$ otherwise