Originally Posted by

**rebghb** Hello everyone!

I'm trying to practice on the **numerous **steps one has to follow to obtain a power series solution so I tried to solve the simple ODE

$\displaystyle (E): y''-2y'+y=0$ whos solutions are very well known to be $\displaystyle y_1=e^x$ and $\displaystyle y_2=x\cdot e^x$.

Happily, I laid down the buliding blocks substituting in $\displaystyle (E), \ y=\sum_{n=0}^\infty c_n \dot x^n$ and, after reindexing and grouping of terms, got the following recurrence relation:

$\displaystyle c_{k+2}=\frac{2(k+1) c_{k+1}-c_k}{(k+2)(k+1)}$.

Okay, I tried to continue by assuming **once**, that $\displaystyle c_0 = 0$ and again $\displaystyle c_1=0$ but I got no pattern at all, I was expecting something **familiar - a taylor series expansion at 0.**

**Question:** (1) When to consider 2 cases, a null $\displaystyle c_0$ and a real $\displaystyle c_1$ and the opposite.

(2) Why didn't I get a pattern?

Thanks!