Originally Posted by

**rebghb** Hello everyone!

I'm studying for power series solution for ODEs and I came over this example:

$\displaystyle y''-(1+x)\,y=0$. And by assuming a power series centered at the ordinary point $\displaystyle x_0=0$ we're bound to get two equalities:

$\displaystyle c_2=\frac{1}{2}\,c_0$ and $\displaystyle c_{k+2}=\frac{c_k+c_{k-1}}{(k+1)(k+2)}$ for $\displaystyle k=1,2,3...$

I went right ahead to continue working the solutions out but I accidently looked at the solution and it shows to sets of calculations: one assuming $\displaystyle c_0=0 \mbox{ and} \ c_1 \mbox{ different than } \ 0$ and another one supposing the opposite.

Can someone explain why is this assumption valid?

Thanks!!