On boundary value problem for equations with cubic nonlinearity and step-wise coefficient

The differential equation with cubic nonlinearity x′′ = −ax + bx3 is considered together with the boundary conditions x(−1) = x(1) = 0 . In the autonomous case, b = const > 0 , the exact number of solutions for the boundary value problem is given. For nonautonomous case, where b = β(t) is a step-wise function, the existence of additional solutions is detected. The reasons for such behaviour are revealed. The example considered in this paper is supplemented by a number of visualizations.