### Home > APCALC > Chapter 10 > Lesson 10.3.2 > Problem10-144

10-144.

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…”

... the product of ___ and ___.

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

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

By partial fraction decomposition:

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

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

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

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

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

Solve for the value of $A$:

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