homogeneous linear differential equations

**1. The problem statement, all variables and given/known data**

The equation

2y'' - y' + y2(1 - y) = 0;

where y' = dy/dx

and y'' = d^2y/dx^2

represents a special case of an equation used as a model

for nerve conduction, and describes the shape of a wave of electrical activity

transmitted along a nerve fibre.

**2. Relevant equations**

the task is to find a value for the constant "a" so that y = [1 + e^(ax)]^(-1) represents the solution of the equation

**3. The attempt at a solution**

y'=-a*e^(a*x)/[e^(a*x)+1]^2

y''=a^(2)*e^(a*x)[e^(a*x)-1]/[e^(a*x)+1]^3

Basically after I substitute the solution into

the equation and did some calculation I've reached this point

[2*(a^2)*e^(a*x)]*[e^(a*x)-1]+[a*e^(a*x)]*[e^(a*x)+1]-e^(a*x)/[e^(a*x)+1]^3=0=>

=>2*(a^2)*(e^(a*x)-1)+a*(e^(a*x)+1)-1=0...any advice?