1. ## Is A small?

I am working through a derivation in relativity and find the statement that A (unit is 1/distance) is small. I don't see how we can make such a statement when A could have the unit $\displaystyle m^{-1}$ or equally the unit could be $\displaystyle light years^{-1}$.

Some background:
The equation that we are solving is:

$\displaystyle \frac{d^2u}{d \phi^2}+u-\frac \alpha 2 = \frac 32 \beta u^2$

We propose a solution of the form:
$\displaystyle u=\frac 1 \rho = A(1+\epsilon cos(\phi-\phi_0))$

Where $\displaystyle \rho$ is the distance between a planet and its sun, A and $\displaystyle \epsilon$ are constants. $\displaystyle \phi_0$ is a slowly varying function of $\displaystyle A\phi$. Clearly A could be, numerically, small if we use light years as the unit of measure.

Does the given DE tell me that u (and hence A) is small?

2. Not sure if this is what you're after but surely since we're talking distance between planets and suns then 1/p is going to be tiny.

I think the statement that A is small is meant as a guideline so that if you get A=a billion then clearly somethings gone wrong...

3. A typical distance between a star and its planets is on the order of $\displaystyle 10^{-7}$ light years so, no, A is not small at all!

4. The motivation for saying that A is small is to allow terms in A^3 to be crossed out (because if A is small then A^3 is very small) of a complicated equation allowing an acurate aproximation to be found.