I am dealing with the energy functional of a given physical system and I want to find the values of x that which makes it extreme. Its expression is given by
E[x_,a_,b_]:=(pi/2)*x^2-(pi/4b)*a*x^2-(pi/2)*a*PolyLog[2,-(1/2b)*x^2]
so when we differentiate it with respect to x, we get
dE[x_,a_,b_]:=pi*x-(pi/2b)*a*x+pi*(a/x)*Log[1+(1/2b)*x^2]
So the problem comes down to finding the values of x that make dE equal to zero. I have tried to use Solve and Reduce, but it hasn't taken me anywhere. For instance, what I have tried so far are things like
Reduce[(1 - a/b)*x^2 + a*Log[1 + x^2/b] == 0, x](*this come from setting dE==0*)
or
Solve[(1 - a/b)*x^2 + a*Log[1 + x^2/b] == 0, x]
The former takes a lot of time to be computed and I always end up aborting the evaluation while the latter yields the following:
{{x -> -(Sqrt[-b^2 + b a + b a ProductLog[(E^(-1 + b/a) (b - a))/a]]/Sqrt[b - a])}, {x -> Sqrt[-b^2 + b a + b a ProductLog[(E^(-1 + b/a) (b - a))/a]]/Sqrt[b - a]}}
In principle, I can constrain the range in which the parameters a and b are allowed to vary. Ideally, these would be `
0<a<3.2*10^10 && 0<b<1000
I would really appreciate if someone could give me a hint on how to get these solutions for x.