I'm trying to solve the following ODE in $x\in[0,R]$:
$\frac{f''(x)x^2+f'(x)x + (x^2-1)\sin(f(x))\cos(f(x))}{x}=0 \quad$ and $f(0)=0$ and $f'(R)=0$.
I tried several stuff by hand but also asked Maple and WolframAlpha but both of them do not give any answer.
I.e. I found out that the solution of $f''(x)+\sin(f(x))\cos(f(x))=0 $ is $f(x)= \text{JacobiAM}(x/k,k)$ since $\frac{\partial^2 \text{JacobiAM}(x,k)}{\partial x^2} = -k^2\sin(f(x))\cos(f(x))$ where $\text{JacobiAM}(x,k)$ is the Jacobi Amplitude Function Wikipedia/Jacobi_elliptic_functions.
Furthermore, I obtained numerical solutions in terms of power series and Fourier series. The qualitative form of the solutions are quite similar to a Jacobi Amplitude Function.
My question therefore is: Is it hopeless to find a closed form solution for this ODE?
I also thought about transforming the ODE using Fourier transforms. If the transform is inside trigonometric function this does not really help or does it?
Do i miss i particular point or is this ODE really that problematic?
The function does not look very spectacular:
I solved f(x) with $R=2\sqrt{2}$ using a Power series approximation up to $x^6$ :
($\rho$ is $x$)
I kindly ask to provide me a hint for a method which i also could try out to obtain e.g. a series solution or in terms of a transformation.