Chapter 7.2, Problem 11CP

Solve

$\{\begin{array}{c}{y^{\prime }}^{\prime }=cy\left(1-y\right)\\ y\left(0\right)=0\\ y\left(1/2\right)=1/4\\ y\left(1\right)=1\end{array}$

for

$c>0$, within three correct decimal places. (Hint: Consider the BVP formed by fixing two of the three boundary conditions. Let $G\left(c\right)$ be the discrepancy at the third boundary condition, and use the Bisection Method to solve $G\left(c\right)=0$. )