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.
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!