Initial Value Problem, confused due to non-linearity

In summary, the given equation is a special case used to model nerve conduction and describes the shape of a wave of electrical activity transmitted along a nerve fibre. The question asks to find a value of the constant a for which the given solution is a solution of the equation. The solution is found to be a=-0.5 after trial and error. The poster also mentions that this is not an exam question and they have already solved it.
  • #1
TaliskerBA
26
0

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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  • #2
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?
 
  • #3
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.
 

FAQ: Initial Value Problem, confused due to non-linearity

What is an initial value problem?

An initial value problem is a type of differential equation that involves finding a function or curve that satisfies certain conditions at a specific point in time. These conditions usually include an initial value, such as the value of the function at time zero, and a differential equation that describes the rate of change of the function over time.

How does non-linearity affect an initial value problem?

Non-linearity refers to a situation where the rate of change of a function is not directly proportional to the function itself. This can complicate the solution of an initial value problem, as the usual methods for solving linear equations may not work. Non-linearity can also lead to multiple possible solutions or chaotic behavior.

What are some common examples of non-linear initial value problems?

Non-linear initial value problems can be found in many fields of science and engineering, such as physics, biology, and economics. Some common examples include the Lotka-Volterra equations for predator-prey interactions, the logistic growth model, and the heat equation in thermodynamics.

How can non-linearity be handled in solving an initial value problem?

There are several methods for solving non-linear initial value problems, including numerical methods, series expansions, and perturbation methods. These techniques involve approximating the solution using a series of simpler equations or iterating towards a solution. The specific method used will depend on the complexity of the problem and the desired level of accuracy.

Can non-linear initial value problems be solved analytically?

In many cases, non-linear initial value problems cannot be solved analytically, meaning that there is no exact, closed-form solution. Instead, numerical or approximation methods must be used to find an approximate solution. However, there are some special cases where analytical solutions can be found, such as for certain types of non-linear equations with only a few variables.

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