Solving a nonlinear first order differential equation

[Chromatic_Universe]

(a'[t]/a[t])^2 == K*(A + B*a[t]^-6)^1/2} is the equation to be solved for getting the solution of a(t) in terms of time(t). Any ideas on how to solve this problem? Use of Matlab or Mathematica is accepted.

 

[Orodruin]

Are you after an analytic or a numerical solution? The numerical solution should be rather straight-forward in both Matlab and Mathematica. Just insert the differential equation along with your initial conditions into the appropriate differential equation solver.

 

[Chromatic_Universe]

Are you after an analytic or a numerical solution? The numerical solution should be rather straight-forward in both Matlab and Mathematica. Just insert the differential equation along with your initial conditions into the appropriate differential equation solver.

I am looking for an analytical solution. Sorry for missing that out.

 

[Orodruin]

Your differential equation is separable. You could just solve for $dt$ in terms of $da$ and $a$ and integrate. However, doing so will likely result in a hypergeometric function.

 

[Chromatic_Universe]

Your differential equation is separable. You could just solve for $dt$ in terms of $da$ and $a$ and integrate. However, doing so will likely result in a hypergeometric function.

I do get a Hypergeometric solution while solving in Mathematica, but is there any physical implication of this solution?

 

[phyzguy]

I do get a Hypergeometric solution while solving in Mathematica, but is there any physical implication of this solution?

Since we don't know what a(t) represents, how could we know the physical implications?

 

[Chromatic_Universe]

Since we don't know what a(t) represents, how could we know the physical implications?

So a(t) is the scale factor in the FLRW metric in Cosmology https://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric, the functional form of a(t) shows how the Universe behaves(expanding/collapsing) and the rate of such behaviour.

 

[Orodruin]

Well, yes, the scale factor’s behaviour with time gives you some physical information. It is usually easier to consider the behaviour of the derivatives if you want to know how your spacetime behaves.

 

[Chromatic_Universe]

Well, yes, the scale factor’s behaviour with time gives you some physical information. It is usually easier to consider the behaviour of the derivatives if you want to know how your spacetime behaves.

I was thinking what is physically implicated by the hypergeometric behaviour of the a(t).