Can you suggest a software to solve this equation? Can you please suggest a free software or website that would allow me to approximate, numerically, the first $n$ roots of the equation $ 2J_{0}(2\alpha)+2\alpha J'_{0}(2\alpha)=0$ ? I'm trying to find a Bessel-Fourier expansion for a function in the interval $[0,2]$ and that's the boundary condition given. As you know, I need the roots of that equation to find the eigenvalues. I've checked Maxima and Octave but I think they don't offer this-or at least it wasn't clear to me they had this feature. I need this ASAP, Thanks in advance!
Note: By $J_{0}$ we mean the Bessel function of order $0$
 A: The $n^{\text{th}}$ zero of function $$f(x)=J_0(2 x)-2 x J_1(2 x)$$ is quite close to
$$x^{(n)}_0= -1+ \frac \pi 2 n$$
Performing one iteration of Halley method
$$x^{(n)}_1=\frac t2\,\,\frac{2 t \left(t^2+2\right) J_0(t){}^2-\left(t^2-1\right) J_1(t) \,J_0(t)+t
   \left(t^2+1\right) J_1(t){}^2 } {2 t\left(t^2+1\right) J_0(t){}^2+\left(t^2-1\right) J_1(t)\, J_0(t)+t \left(t^2+3\right)
   J_1(t){}^2 }$$ where $t=(\pi\,n-2)$.
For example
$x^{(123)}_1=192.0347$ while the solution is $192.0307$.
For sure, better could be done using instead Householder method (the formula is too long to be typed here). For the above case, the result would be $192.0304$.
Tabulating once the values of $J_0(t)$ and $J_1(t)$ could make life quite easy except if you are looking for very accurate results.
A: Javier Segura has written a Fortran implementation of his 2012 work with Amparo Gil concerning the roots of $x \mathcal{C}'_{\nu}(x)+\gamma \mathcal{C}_{\nu}(x)$, $\gamma\in\mathbb{R}$, $\mathcal{C}$ a linear combination of Bessel functions.
Gil, Amparo, and Javier Segura. 2012. “Computing the Real Zeros of Cylinder Functions and the Roots of the Equation $x\mathcal{C}_{\nu}′(x)+\gamma \mathcal{C}_{\nu}(x)=0$.” Computers & Mathematics with Applications 64 (1): 11–21.
