Finding a specific solution to an ODE system

In the code I posted, NDSolveValue creates interpolating functions for both o and r. Actually, because I asked it for o[t] and r[t], rather than o and r, the argument t is included in the output. If you execute the NDSolveValue without the semicolon you will see two interpolating functions, each expressed as a function of t. They are assigned to the variables fo and fr.

The first time at which rLimit is crossed is tList[[1,1]], so the value of o at that time is just fo/.t->tList[[1,1]] (Look up Rule and Replace.)

In[10]:= fo /. t -> tList[[1, 1]]

Out[10]= 20.0634

You can of course plot fo[t] just the way I plotted fr[t]. You could even put them both in the same plot.

Best, David

It would be easier to respond if you would post complete code or a new notebook. However, if you are not interested in the values for r after it first passes a limit, you could add WhenEvent to the NDSolve equations. In the code below I used your intial notebook and set rLimit to 2.5. WhenEvent stops integration when r hits 2.5, and records the time in tStop. So r is then r[tStop]. (Note that, for some reason, tStop needs to be defined prior to executing NDSolve or the value will not be saved.)

In[1]:= L = 2 Pi;
M = 1;
x = 35 Degree;

In[18]:= rLimit = 2.5; tStop = 0;

In[19]:= TotalThrust = 
  NDSolve[{r''[t] == - (4*Pi^2 / r[t]^2) + ((o'[t]^2)*r[t]) + 
      HeavisideTheta[5 - t]*0.51*Cos[x], 
    o''[t] == (-2 r'[t]*o'[t])/r[t] + 
      HeavisideTheta[5 - t]*0.51*Sin[x], o[0] == 0, o'[0] == 2*Pi, 
    r[0] == 1, r'[0] == 0, 
    WhenEvent[r[t] >= rLimit, {"StopIntegration", tStop = t}]}, {o [
     t], r[t]}, {t, -10, 10}];

In[21]:= tStop

Out[21]= 5.27189

Is there any way to specify the first solution at around t=7

Use FindRoot. FindRoot requires that you give it a starting value to guess from. So you can try something like

FindRoot[First[r[t] /. TotalThrust] == 5.3, {x,7}]

Hello again!

Sorry about the disagreement with the graph, in my currently used model, different initial conditions are set such that r does eventually reach the desired value.

What is the syntax of the Nsolve and Findroot when solving a specific equation like First[r[t] /. TotalThrust] == 5.3?

I've attempted with NSolve [First[r[t] /. TotalThrust] == 5.3, t] and it returns a value of 16.22 which is wrong when looking at my graph, and returns the error : Inverse functions are being used by NSolve, so some solutions may not be found; use Reduce for complete solution information. >>

Thanks!