Am I doing this correctly please?
If we have:
$$ \frac{d^2y}{dx^2} + y = \frac{1}{a + \epsilon y} $$
where y is a function of x.
If i need to rescale such that $ y(x) = \alpha z(x) $, is the following the correct procedure?
We substitute in for $ y(x)$ to give:
$$ \begin{align} \alpha \frac{d^2z}{dx^2} + \alpha z = \frac{1}{a + \epsilon \alpha z} \\ \frac{d^2z}{dx^2} + z = \frac{1}{a \alpha + \epsilon \alpha^2 z} \\ \frac{d^2z}{dx^2} + z = \frac{1}{1 + \delta z} \end{align} $$
Where $ \alpha = \frac{1}{a} $, $ \delta = \epsilon \alpha^2 = \frac{\epsilon}{a^2} $
Just looking for a Yes or No please, and if No a pointer as to what I may be doing wrong.
Thankyou in advance.
Yes, if you know that $a \neq 0$.