## Potential series method

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

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hi matematikuvol!
 Quote by 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)

 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.

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## Potential series method

 Quote by 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.
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