For the differential equation $\frac { d y } { d x } = 2 y ( 1 - y )$:

1. What type of growth is described by this equation? Finish the sentence, “The rate of growth is proportional to…”

   Hint (a):

   ... the product of ___ and ___.

2. Solve the differential equation with initial condition $y(0) = 0.5$ and sketch the solution curve.

   Step 1 (b):

   $\frac{dy}{2y(1-y)}=dx$

   Step 2 (b):

   By partial fraction decomposition:

   $\int\frac{dy}{2y(1-y)}=\frac{1}{2}\int\Big(\frac{1}{y}+\frac{1}{1-y}\Big)$

   Step 3 (b):

   $\frac{1}{2}\Big(\ln|y|-\ln\Big|\frac{1}{1-y}\Big|\Big)=x+C$

   Step 4 (b):

   $\ln\Big|\frac{y}{1-y}\Big|=2x+C$

   Step 5 (b):

   $\frac{y}{1-y}=Ae^{2x}$

   Step 6 (b):

   $y=\frac{Ae^{2x}}{1+Ae^{2x}}$

   Step 7 (b):

   Solve for the value of $A$:

   $0.5=\frac{A}{1+A}$