Given the equation:
(x^2)y''+xy'-y=0
How do you write this in power series?
The DE equation ...
$\displaystyle x^{2}\cdot y^{''} + x\cdot y^{'} - y=0$ (1)
... belongs to a family which is called 'Euler's equation' and its particular solutions are of the form $\displaystyle x^{\alpha}$. Substituting in (1) $\displaystyle y= x^{\alpha}$ we obtain that it must be...
$\displaystyle \alpha = \pm 1$ (2)
... so that ther general solution of (1) is...
$\displaystyle y= c_{1}\cdot x + \frac{c_{2}}{x}$ (3)
... which is itself a Laurent series expansion...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$