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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