Potential series method

matematikuvol

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

tiny-tim

hi matematikuvol!
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
[tex]\alpha(x)y''(x)+\beta(x)y'(x)+\gamma(x)y(x)=0[/tex]
if ##\alpha(0)=0## you must work with ##\sum^{\infty}_{n=0}a_nx^{n+k}##, but I'm not sure.

tiny-tim

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