In[]:=
ClearAll["`*"];Remove["`*"]
In[]:=
v[t_]:=v0(1-Exp[-Sqrt[2/3]x[t]/mp]);ode1=1/2*x'[t]^2+a''[t]/a[t]+a'[t]/a[t]^2;ode2=x''[t]+3(a'[t]/a[t])x'[t]+v[t];
In[]:=
DSolve[{Expand[ode1*a[t]^2]==0,Expand[ode2*a[t]]==0},{x[t],a[t]},t](*Can't*)
Out[]=
DSolve
′
a
[t]+
1
2
2
a[t]
2
′
x
[t]
+a[t]
′′
a
[t]0,v0a[t]-
-
2
3
x[t]
mp

v0a[t]+3
′
a
[t]
′
x
[t]+a[t]
′′
x
[t]0,{x[t],a[t]},t
AsymptoticDSolveValue[{Expand[ode1*a[t]^2]==0,Expand[ode2*a[t]]==0},{x[t],a[t]},{t,0,1}](*Tolongtocompute!!!*)
Out[]=
$Aborted
(*Simplerequation.Iweset:*)
In[]:=
Series
-
2
3
x[t]
mp

,{x[t],0,1}
Out[]=
1-
2
3
x[t]
mp
+
2
O[x[t]]
(*andthen:*)
DSolve
′
a
[t]+
1
2
2
a[t]
2
′
x
[t]
+a[t]
′′
a
[t]0,v0a[t]-1-
2
3
x[t]
mp
*v0a[t]+3
′
a
[t]
′
x
[t]+a[t]
′′
x
[t]0,{x[t],a[t]},t(*Can't*)
Out[]=
DSolve
′
a
[t]+
1
2
2
a[t]
2
′
x
[t]
+a[t]
′′
a
[t]0,v0a[t]-v0a[t]1-
2
3
x[t]
mp
+3
′
a
[t]
′
x
[t]+a[t]
′′
x
[t]0,{x[t],a[t]},t
In[]:=
AsymptoticDSolveValue
′
a
[t]+
1
2
2
a[t]
2
′
x
[t]
+a[t]
′′
a
[t]0,v0a[t]-1-
2
3
x[t]
mp
*v0a[t]+3
′
a
[t]
′
x
[t]+a[t]
′′
x
[t]0,{x[t],a[t]},{t,0,1}(*Tolongtocompute!!!*)
Out[]=
$Aborted