# Differential equation for function involving hyperbolic Sine

Let $y(z) = 2 \frac{sinh^{-1}(z/(2a)^{1/2})}{(z^2 +2a)^{1/2}}$

where $sinh(z)$ is the sine hyperbolic function.

$$ K(z):=\frac{1}{z(y(z) - y(-z))} \rightarrow 1/2\,{\frac {a}{{z}^{2}}}+1/6-{\frac {{z}^{2}}{90\,a}}+{\frac {{z}^{4} }{378\,{a}^{2}}}-{\frac {23\,{z}^{6}}{28350\,{a}^{3}}}+{\frac {263\,{z }^{8}}{935550\,{a}^{4}}}-{\frac {133787\,{z}^{10}}{1277025750\,{a}^{5} }}+{\frac {157009\,{z}^{12}}{3831077250\,{a}^{6}}}-{\frac {16215071\,{ z}^{14}}{976924698750\,{a}^{7}}}+{\frac {2689453969\,{z}^{16}}{ 389792954801250\,{a}^{8}}}+O \left( {z}^{18} \right) $$

How can I find the linear differential equation in $\frac{\partial}{\partial z }$ with coefficent in the polynomial ring $\mathbb{C}[z]$ that annihilates $K(z)$. I am unable to do it by hand I think some software in sagemath might help.

In your definition :

$$K(z) := \frac{1}{z(y(z)-y(-z))}$$

I can't make sense of the denominator, where $z$ is simultaneously a function and a dependent variable... unleess you have meant :

$$K(z) := \frac{1}{(y(z)-y(-z))\cdot z}$$

Please clarify !

BTW...

Is that homework ?

Isn't $y(z)$ an odd function with $y(z)-y(-z)=2y(z)$?

Yes I have meant what you wrote. No it's homework question, it's part of a question I ask in last post, which I cannot solve. So I trying to break it in parts and analyse.

The last post https://ask.sagemath.org/question/623... I realise that if there is a recursion in the coefficient of $K(z)$ that might help me to solve my question.