Multiple positive solutions for the Dirichlet boundary value problems by phase plane analysis (Q304907)

Summary: We consider boundary value problems for the scalar differential equation \[ x''+ \lambda f(x)=0, \] subject to the boundary conditions \[ x(0)=0, x(1)=0, \] where \(f(x)\) is a seventh-degree polynomial and \(\lambda\) is 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.