# differential equations – Solving a non-linear boundary value problem for a small non-linearity

I am trying to solve the differential equation
$$frac{d^2y}{dx^2}-frac{a}{b}y-frac{u}{b}y^3=0,quad frac{dy}{dx}Bigvert_{x=0}+frac{s}{b}yvert_{x=0}=0,quad frac{dy}{dx}Bigvert_{x=L}-frac{s}{b}yvert_{x=L}=0.$$ For $$u=0$$, and $$Ltoinfty$$, the homogeneous system allows a non-trivial solution only for $$a=frac{s^2}{b}$$. Furthermore, this critical value of $$a$$ increases with decreasing $$L$$. I am trying to solve the non-linear problem with $$uneq 0$$, while allowing an extrapolation to the limit $$uto 0$$. I am expecting that a solution would show up for $$alessapprox frac{s^2}{b}$$, which is what I am trying to see explicitly.

This script does not seem to be working. Any help on this would be appreciated.

``````{b, s, u, L, a} = {0.05, 0.1, 0.0001, 100, 0.1}; sol =
NDSolve({y''(x) - (a/b)*y(x) - (u/b)*(y(x))^3 == 0,
y'(0) + (s/b)*y(0) == 0, y'(L) - (s/b)*y(L) == 0}, y(x), {x, 0, L});
Plot(Evaluate(y(x) /. sol), {x, 0, L}, PlotRange -> All)
``````