Initial Value Problem, confused due to non-linearity

[TaliskerBA]

Homework Statement

The equation 2y'' - y' + y^2(1-y) = 0
is 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.

Find a value of the constant a so that y = (1 + e^(ax))^(-1) is a solution of this eqaution.

Homework Equations

y' = -(ae^(ax))/(1+e^(ax))^2
y'' = 2(a^2)(e^(2ax))/(1+e^(ax))^3 - (a^2 * e^(ax))/(1+e^(ax))^2

The Attempt at a Solution

I could only attempt trial and error. I tried out different values of a as integers but got nowhere. a=0 produces the result 1/4 = 0.

We have only studied linear equations with constant coefficients and how to solve these using auxiliary equations, but as this equation is non-linear I have no idea how to solve it.

I would really appreciate some help, I have been trying to work this out for about 2 hours now!

Thanks

 

[LCKurtz]

Are you the same poster as AkilMAI? I just responded to this exact question from him above. Big coincidence if you are not the same. Is this some exam question I shouldn't have answered?

 

[TaliskerBA]

No it's not an exam question, he must just go to the same university as me. I worked it out in the meantime anyway, came out with a=-0.5.