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Why equations

[tex]x(1-x)\frac{d^2y}{dx^2}+[\gamma-(\alpha+\beta+1)x]\frac{dy}{dx}-\alpha \beta y(x)=0[/tex]

should be solved by choosing

##y(x)=\sum^{\infty}_{m=0}a_mx^{m+k}##

and not

##y(x)=\sum^{\infty}_{m=0}a_mx^{m}##?

How to know when we need to choose one of the forms.

Also when I sum over ##m##, then ##\sum^{\infty}_{m=0}a_mx^{m+k}=y(x,k)##. Right?

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# Ordinary differential equations. Series method.

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