How to solve a couple of ODEs in two variables

Questions
,
1

Any help how to solve a couple of ODEs to get the variables a[t] and x[t] ?

ode1= -1/2* x’[t]^2 == a’’[t] / a[t] + a’[t] /a[t]^2

And:

ode2 = x’’[t] + 3 (a’ [t]/ a[t] ) x’[t] + v[t] ==0

where

v[t] = v0 ( 1 - Exp [ - Sqrt [2/3] x[t]/mp ] )
2

Many nonlinear ODEs do not have symbolic solutions in terms of commonly defined functions. The ODE in the question probably falls in that category. However, it should be noted that DSolve sometimes is unable to solve ODEs that do have symbolic solutions in terms of commonly defined functions.In your case probably No hope for solution.

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3

Thank you so much for your answer.

I tried to do another thing:

First: To substitute by x'[t] and x''[t] in ode2 in terms of ode1 :

x'[t] := Sqrt[2 (a''[t]/a[t] + a'[t]/a[t]^2)]

x''[t] := D[x'[t], t]

Then

ode2 = x''[t] + 3 a'[t]/a[t] x'[t] + v[t] == 0

Gives:

Out[18]= (1-E^(-((Sqrt[2/3] x[t])/mp)))^2 v0+(3 Sqrt[2] (a^\[Prime])[t] Sqrt[(a^\[Prime])[t]/a[t]^2+(a^\[Prime]\[Prime])[t]/a[t]])/a[t]+(-((2 (a^\[Prime])[t]^2)/a[t]^3)+(a^\[Prime]\[Prime])[t]/a[t]^2-((a^\[Prime])[t] (a^\[Prime]\[Prime])[t])/a[t]^2+(a^(3))[t]/a[t])/(Sqrt[2] Sqrt[(a^\[Prime])[t]/a[t]^2+(a^\[Prime]\[Prime])[t]/a[t]])==0

But now there is in ode2, x[t] which comes from v(t) .. So how to integrate ode1:

x'[t] := Sqrt[2 (a''[t]/a[t] + a'[t]/a[t]^2)]

To get x[t]

Then solve ode2 as a third order differential equation in a'''[t].