Solving 2nd Order Nonlinear DE: Analyzing Effects of g Changes

[DanS]

Hi everyone,

I'm having a hard time analyzing the following problem:

b v(x) = -exp(-x) - 1/2 ( g v'(x) )^2 - n x v'(x) + S(g) v''(x)

where:
v' = dv/dx, etc.
0 < b< 1
g > 0
n > 0
S(g) >0 and S'(g) >0
x \in (-inf, inf)

The main goal is to figure out what happens as g changes. Specifically, I want to know if the cross partial:
( d^2v(x) )/ (dx dg) >= - (dv/dx)/g

For what it's worth, I know that v'>0 and v''<0. Also, v<0 for all x.

I have also written the problem as a 2 dimensional non-autonomous system of differential equations where:
z = v'
y = b v + 1/2 (g v')^2

Then the dynamical system becomes:
z' = ( y + exp(x) + n x z )/S(g)
y' = b z + g^2 z ( y + exp(x) + n x z )/S(g)

as another alternative, a function w(x) = exp(x) can be used to turn this into a 3 dimensional autonomous system.

I don't think this DE can be solved analytically, but any help would be very much appreciated. As I said, I'm specifically interested in what happens as g changes, but I'd also like to prove that v'''>0 (observed in numerical simulations).

Thank you very much in advance,
Dan

 

[jvicens]

Hi Dan,

Thank you for sharing your problem with us. It seems like you have already made some progress in breaking down the problem and finding alternative ways to represent it. This is a good start, as it allows for different approaches to analyzing the system.

I agree with you that this differential equation cannot be solved analytically, so we will need to rely on numerical methods to analyze it. One approach you could take is to use a software program, such as MATLAB or Mathematica, to simulate the system for different values of g and observe the behavior of the solution. This could help you see any patterns or trends as g changes and potentially provide insight into the behavior of the cross partial and v'''.

Another approach is to use mathematical techniques, such as stability analysis, to analyze the behavior of the system. This could also help you understand how the system behaves as g changes. Depending on the specific values of b, n, and S(g), you may be able to make some general statements about the behavior of the system.

In terms of proving that v''' > 0, this may be more challenging without an analytical solution. However, you could try using numerical methods to approximate the third derivative and observe its behavior for different values of g. This could give you some evidence to support your observation from simulations.

I hope this helps and good luck with your analysis! Let me know if you have any further questions.