# Method of Frobenius

Can someone please explain the green text to me?

Maybe I am not reading it right, but that sentence makes no sense to me. "that (2) is an equation For which $xp(x)$ and $x^2q(x)$ " what does this part mean? The next part that follows sys something about being constants. Are they trying to say we should replace equation (2) with $xp(x)$ and $x^2q(x)$ like

$$y''(x) + xp(x)y'(x) + x^2q(x)y(x) = 0$$ So that

$$y''(x) + p_0y'(x) + q_0y(x) = 0$$

Either case it makes no sense to me at all, especially why they started writing out the $xp(x)$ as a series.

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It makes perfect sense to me! –  Mercy Jun 23 '12 at 23:21

Right below Eq. (2) it is said that $p(x)=p_0/x, \ q(x)=q_0/x^2$ which means $xp(x)$ and $x^2q(x)$ are constants. Later on he assumes that these functions are not constants!
Why don't they say instead "that (2) is an equation for $p_0$ and $q_0$, instead of being constants, are analytic functions" –  Hawk Jun 23 '12 at 23:24
@jak Because (2) is an equation for $y$ (which is unknown), not for $p_0$ and $q_0$ (which are considered to be given). The sentence in green would be clearer if it said "equation in which..." –  user31373 Jun 23 '12 at 23:39