An Analytic Solution of the Nonlinear Differential Equation Δ λ = f (λ)

A spherically symmetric solution of the equation Δ\(\lambda\) = f (λ) with the boundary conditions \(\lambda\)→0 as r→ \(\infty \) and (d\(\lambda\)/dr)r=a=a given value, has been obtained under the assumptions that f (\(\lambda\)) is monotonic and can be expanded in a power series about \(\lambda\)=0 and f(0)=0. The series solution λ=\(_{Σ}^\infty\)\(\propto\)^sb_s(r). r^-1, where. \(\propto\) is a parameter independent of r; b₁=A exp (-Kr); K²=f'(o); bs=k^-1\(\int\limits_r^\infty\) \(_{Gs(X)}^{s=1}\) sinh K. (x-r) dx;  Gs= (r/s!). \(\delta^s/\delta\propto^s\)[f (\(\lambda\))-K2\(\lambda\)] \(\propto\) =0 has been proved to be regular in the domain r>0. A solution in the form of an integral equation, namely, r\(\lambda\)(r)=B exp (-Kr)-{-K^-\(\int\limits_r^\infty\)x [f(\(\lambda\))-K²\(\lambda\)] sinh K (x—r) dx has also been derived. Using the integral equation and the series solution the values of \(\lambda\) (r) have been calculated for the particular case Δ\(\lambda\)=K² [exp (w\(\lambda\))—exp (-z\(\lambda\))]/(z+w); (K², w, z are known real constants).