We consider boundary value problems for scalar differential equation $x^{\mathrm{\prime \prime }}+\lambda f\left(x\right)=0$, $x(0)=0$, $x(1)=0$, 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.