Initial Value Problem, confused due to non-linearity

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SUMMARY

The discussion centers on solving the non-linear differential equation 2y'' - y' + y^2(1-y) = 0, which models nerve conduction. The solution y = (1 + e^(ax))^(-1) requires finding the constant a. After attempts with various integer values, the correct solution was determined to be a = -0.5. This highlights the challenges of solving non-linear equations compared to linear ones.

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TaliskerBA
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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
 
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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?
 
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.
 

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