Alternative form of $y = \frac{1}{1 - \left(\frac{x}{a}\right)^2}$?

Question

I solved the differential equation $y'(x) = \frac{2}{a^2}xy^2(x)$ (with $y(0)=1$ and $x\in(-|a|,|a|)$) and came to $$ y(x) = \frac{1}{1 - \left(\frac{x}{a}\right)^2}. $$ This looks like some trigonometric thing to me. Do you know a better form for $y$?

Answer 1

You might think of $$ \tanh'(x)=\frac1{1-x^2}. $$ So you get $$ y(x)=\frac{1}{1-\left(\frac{x}a\right)^2}=\tanh'\left(\frac{x}a\right) $$