I have an operator given by $$K=\cosh ax\hspace{2pt}\partial_t-\frac{\sinh ax}{at}\partial_{x}$$ and my function is $$f(t,x)=e^{ikx}J_{\pm\frac{ik}{a}}(mt)$$ where $J_{\nu}(t)$ is the first kind Bessel function.
Now to check whether it's an eigenfunction all I have to show is $$K\hspace{3pt}f(t,x)=\alpha\hspace{3pt}f(t,x)$$ But what I am getting in LHS is $$e^{ikx}\Big(\cosh ax\hspace{4pt}\partial_{t}\bigg[J_{\pm\frac{ik}{a}}(mt)\bigg]-ik\frac{\sinh ax}{at}\hspace{2pt}J_{\pm\frac{ik}{a}}(mt)\Big)$$ If I differentiate the Bessel function I am unable to write it in closed form i.e. $$\partial_xJ_{\nu}(x)=\frac{1}{2}\Bigg[\frac{\nu}{\Gamma(\nu +1)}\Big(\frac{x}{2}\Big)^{\nu-1}-\sum_{s=0}^{\infty}\frac{(-1)^s}{m!\Gamma(\nu +s+1)}\Big(\frac{x}{2}\Big)^{\nu+2s}\bigg(\frac{1}{m+1}+\frac{1}{\nu +m+1}\bigg)\Bigg]$$ How should I proceed? I know that $\alpha$ has to be of form $i\beta$.