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].