I'm trying to solve $\frac{d^2\varphi}{ds^2}=\lambda^{2}\sin \varphi$ using perturbation methods which in non dimensional form can be written as $\frac{d^2\varphi}{d\xi^2}=\sin \varphi$.
With approximations as: $\varphi(\xi)\approx\varphi_0(\xi)+\epsilon\varphi_1(\xi)+...$ and $\sin\varphi\approx\varphi+...$ the leading order yields $\varphi=c_1e^\xi+c_2e^{-\xi}$ and the first order terms are then cancelled out (assuming that the boundary conditions at 0 are known). The solution then is highly divergent. How should I proceed in order to get something "solvable" by hand? Does it make sense to try solve the initial form including dimensions?
FYI the problem is about capilary rise hence, $\lambda^2=\rho g/\gamma$ such that $[\lambda]=L^{-1}$
Thank you very much for your help!