In[]:=
$Version
Out[]=
14.1.0 for Microsoft Windows (64-bit) (July 16, 2024)
In[]:=
ClearAll["`*"];
In[]:=
DSolve[{a*ϕ1''[t]+b*(ϕ1''[t]+ϕ2''[t])+c*Sin[ϕ1[t]]==0,d*ϕ2''[t]+e*Sin[ϕ2[t]]==0},{ϕ1[t],ϕ2[t]},t](*AnalyticallyCan'tsolve*)
DSolve
:Inverse functions are being used by DSolve, so some solutions may not be found.
Out[]=
DSolve[{cSin[ϕ1[t]]+a
′′
ϕ1
[t]+b(
′′
ϕ1
[t]+
′′
ϕ2
[t])0,eSin[ϕ2[t]]+d
′′
ϕ2
[t]0},{ϕ1[t],ϕ2[t]},t]
In[]:=
sol=With[{a=1,b=2,c=3,d=1/2,e=1},NDSolve[{a*ϕ1''[t]+b*(ϕ1''[t]+ϕ2''[t])+c*Sin[ϕ1[t]]==0,d*ϕ2''[t]+e*Sin[ϕ2[t]]==0,ϕ1[0]==0,ϕ1'[0]==Pi/2,ϕ2[Pi/9]==2Pi,ϕ2'[0]==Pi},{ϕ1,ϕ2},{t,0,2*Pi}]]
Out[]=
ϕ1InterpolatingFunction
Domain: {{0.,6.28}}
Output: scalar
,ϕ2InterpolatingFunction
Domain: {{0.,6.28}}
Output: scalar

In[]:=
Plot[Evaluate[{ϕ1[t],ϕ2[t]}/.sol],{t,0,2Pi},PlotLegends{"ϕ1[t]","ϕ2[t]"}]
Out[]=
ϕ1[t]
ϕ2[t]