- F(x) = 1/(1+1/Kx) is not a solution of the differential equation you show. It is a solution of  x d/dx(Fx) = - F(x)(1-F(x)) . You forgot a minus sign.

- The solution of the differential equation you give is 1/(1+Kx)

- F(x) = 1/(1+1/Kx)**2 is a solution of

x d/dx(F(x)) = -2 F(x)(1+1/(Kx) - F(x))

Hope I did not make any mistake.

Cheers,

Henri

Hi Robert,

It can be proved that (1+K/x)^-2  satisfies the equation x d/dx F=2F(1-sqrt(F)).

I am going to send you right away a document that explains the methodology to obtain such differential equation.

Here is the methodology to derive the differential equation satisfied by such function. The attached document is written for the case where the exponent is 2, but it can easily be generalized to any exponent (non null).

Antonin, thank you very much!  Looks very usable for my purpose.  I must now backtrack this into a rather complicated partial differential equation with tensors.  It'll take me a couple of days probably. 

The general solution for any exponent that you gave should be quite useful also, allowing me to define a general family of tensor PDEs with certain properties.  If you are interested I'll follow up with you later.

dF/dx= 2K sqrt(F)(sqrt(F)-1)2

 Hi Robert,

The generalization to any non null exponent alpha (see figure function.png) is the equation given in figure equation.png. 

I am very interested by the way you are going to apply this in your research. Please send us a message by adding an answer on this topic once you come up with something.

Best.

The solution seems complete - there is nothing to be added. For my own curiosity I would like to check if the equations are in the handbooks : Kamke or Polyanin.

A small question : what about the solutions of the equation xF'(x) = ... around x=0 ?

Let y = F(x), y' = dF/dx = dy/dx. Then, just "differentiate, and see what you get". I obtain the following equation:

dy/dx =( 2 times y to 3/2)/(K. x squared). 

Vladimir, the equation I was asking about comes from the Schwarzschild derivation, see link below.  Instead of A and r, I substituted F and x to be general.

So the solutions near x=0 correspond to the solutions near the singularity of a black hole, and have been controversial at least since the 1940s as to whether they are physically real or not.

The solutions near x=-K correspond to solutions near the event horizon of a black hole, and are nearly as controversial.  Not that they aren't widely accepted (though a few object), just that there are controversies over exactly what they imply, as for example the debate over entropy between Susskind and Hawking.

When I went to backtrack to see how the differential equation with solution having a -2 rather than a -1 exponent would affect the field equation of general relativity, I was quite taken by surprise.  The Schwarzschild derivation does not use the whole field equation, only the "vacuum equation" which requires the Ricci tensor to vanish.  So my alternate solution would require the Ricci tensor to not vanish.  I'm still looking into exactly what this means.  Possibly that there would have to be non-local effects?  That would explain why Einstein was so against non-locality.  The physical mechanism I postulated in a 2013 paper for one of the alternate solutions involved uncertainty of position and momentum, so definitely some non-locality.

https://en.wikipedia.org/wiki/Deriving_the_Schwarzschild_solution

Obiously the point is to get the equation for which K is the integration constant and does not appear as a coefficient in the equation. For any function F=f(x, K) you can find equation to which it is general solution with K  being the integration constant. Simply solve with respect to K=K(x, F) and apply dK=0. Best, P

P. Kosinski, but then the new K is not constant, so the original form is inapplicable.  In fact, for the solution I'm looking for of form (1+K/x)-2 the Ricci tensor cannot be everywhere equal to zero.  This can be determined by just substituting it back into the Ricci tensor and doing the algebra.

If I understand you, you are saying to assume a solution of the form (1+K(x)/x)-2.  For a moment that appears interesting, until one realizes that most likely such a solution would simplify to (1+K/x)-1.  I think I just have to face the fact that for my desired solution, the Ricci tensor differs from zero, and explain and justify that.

Just take the total derivative of the solution, and you will have the differential equation: F ' = -2 (1+k/x)^(-3) (-k/x^2)

The question can be reformulated as: Given a parametrized family of curves what is the differential equation of the family. The general answer is that you take as many derivaives as the number of parameters then eliminate the parameter from the arising system of equations. The result is as seen in the attached file:

-4 y[x]^2 + 4 y[x]^3 + 4 x y[x] y′[x] ⩵ x^2 y′[x]^2

Best, János

Dear Robert, no, K is just a constant by the very construction. Best, P

Here k in not constant  the given form cannot be applied.

Dear Robert, if I understand well, you look for a function G(x) such that dG/dx equals your F(x)=(1+1/(Kx))^-2 (did I quote correctly your 1/Kx ?). I use often Maple to find analytical solutions (if any) of integral/differential equations. In this case the software has provided a general solution, shown in attachment. Of course it does not guarantee that this solution fits your boundary conditions, not mentioned in the question. Seb

y=(1+K/x)^-2  satisfies the following 1st-order ODEs

y'=2K*x^{-2}*y^{1.5}

y'=(2/K)*(y^{0.25}-y^{0.75})^2

Also it satisfies the 2nd-order ODE

y*y''=(1.5-2x^{-1}*y)*y'

etc.

There seems to be some continuing interest in my little problem beyond the first or even second or third correct answer.  Soooo... how about this challenge.  Take an answer you feel is correct, and back substitute into the equations from whence the original DE was deduced, and from there back into the vacuum equation (Ricci tensor = 0) from whence those equations came, and show what is the exact expression for the non-zero Ricci tensor that leads to these solutions?  Can you do that?  If you can, I'll be happy to help you write it up and I'm pretty sure it'll be published and noticed.

The problem should be specified, namely  a sense of the  constant k ought to be explaine - if it is a free parameter appering owing to the Cauchy initial data, and there are  no other cosntants, this means that equation under question should be of thefirst order. If so, it can be found a very simople way - it is enough to finf this constant k from the solution presented above and differentiate it with respect to the independent variable x:

one obtains easily that 

k=(((F(x)^{-1/2}-1)⁻¹)/x),

whence we get finally that the diffrenetial equation under search looks as

((dx)/(dx))[(((F(x)^{-1/2}-1)⁻¹)/x)]=0.

What is interesting, this equation is a uniqely possible whiсh possesses the solution F(x) =(1+1/kx)-2!

Regards!

It is indeed correct that you should eliminate the free parameter. Rewrite the original as (y^(-1/2)-1)x=1/k and take the total derivative. Then rearrange to your liking.

John Starrett@

Yet it depends, as I mentioned before,  on the Cauchy data nature  of this parameter ... otherwise, the problem becomes  not uniquely solved, and  even    senseless..

Regards!

OK, thanks all.  Moving on, I have posted a phase 2 question.  See link below.

https://www.researchgate.net/post/What_is_the_non-zero_Ricci_tensor_equation_for_a_metric_field_slightly_non-Schwarzschild