The differential equation with cubic nonlinearity x = −ax + bx 3 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.

We consider boundary value problems for scalar differential equationx′′+λfx=0,x(0)=0,x(1)=0, wheref(x)is a seventh-degree polynomial andλis a parameter. We use the phase plane method combined with evaluations of time-map functions and make conclusions on the number of positive solutions. Bifurcation diagrams are constructed and examples are considered illustrating the bifurcation processes.

No abstract

The nonlinear oscillators describing by differential equations of the form x 00 = −ax + cx5 and x 00 = −ax + bx3 + cx5 are studied. Multiplicity results for both types of equations, given with the Neumann boundary conditions are obtained. It is shown that the number of solutions depend on the coefficient a only. The exact estimates of the number of solutions are obtained. Practical issues, such as the representation of solutions in terms of Jacobian elliptic functions and calculation of the initial values for solutions of boundary value problems, are considered also. The illustrative examples are provided. Outlines of future research conclude the article.

Two-point boundary value problems for second-order ordinary differential equations of Lie´nard type are studied. A comparison is made between equations x´´ + f (x) x´2 + g(x) = 0 and x´´ + f (x) x´ + g(x) = 0. In our approach, the Dirichlet boundary conditions are considered. The estimates of the number of solutions in both cases are obtained. These estimates are based on considering the equation of variations around the trivial solution and some additional assumptions. Examples and visualizations are supplied.