Math Help - 3-Term Recurrence Relation Series Soultion

1. 3-Term Recurrence Relation Series Soultion

Hello everyone!

I'm studying for power series solution for ODEs and I came over this example:
$y''-(1+x)\,y=0$. And by assuming a power series centered at the ordinary point $x_0=0$ we're bound to get two equalities:
$c_2=\frac{1}{2}\,c_0$ and $c_{k+2}=\frac{c_k+c_{k-1}}{(k+1)(k+2)}$ for $k=1,2,3...$

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 $c_0=0 \mbox{ and} \ c_1 \mbox{ different than } \ 0$ and another one supposing the opposite.

Can someone explain why is this assumption valid?

Thanks!!

2. The assumption is valid because the second order ODE is $linear$ and its general solution has the form...

$y(x) = c_{1}\cdot \varphi_{1} (x) + c_{2}\cdot \varphi_{2} (x)$ (1)

... where $\varphi_{1} (x)$ and $\varphi_{2} (x)$ are two particular independent solutions, $c_{1}$ and $c_{2}$ two 'arbitrary constants'...

Kind regards

$\chi$ $\sigma$

3. Originally Posted by rebghb
Hello everyone!

I'm studying for power series solution for ODEs and I came over this example:
$y''-(1+x)\,y=0$. And by assuming a power series centered at the ordinary point $x_0=0$ we're bound to get two equalities:
$c_2=\frac{1}{2}\,c_0$ and $c_{k+2}=\frac{c_k+c_{k-1}}{(k+1)(k+2)}$ for $k=1,2,3...$

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 $c_0=0 \mbox{ and} \ c_1 \mbox{ different than } \ 0$ and another one supposing the opposite.

Can someone explain why is this assumption valid?

Thanks!!
Any initial conditions given?

I believe that the authors wanted to avoid the messy algebra and wanted to isolate each solution separately (Chisigma hinted at that).

4. No it's not an IVP or anything, it's just the concept I am after.
Thanks anyways!