Need to find value of NDSolve at specific value in range

Question

I am working with NDSolve to plot an ODE as a function of time. I need to find a way to figure out the exact value X of the solution for some specific time, say t*. Even better, a way to figure out the time t* at which the F[t*]=X. It's probably straightforward but don't have much experience with NDSolve! Thanks in advance!

Answer 1

You can use WhenEvent. Let's say we want the points that the solution takes the value of 2.5:

{sol,points}=Reap@NDSolve[{D[y[x],x]==Sin[x], y[0]==1, WhenEvent[y[x]==2.5,Sow[{x,y[x]}]]}, {y[x]}, {x, 0, 10}];

pointsis our desired set of points:

{{{2.0944,2.5},{4.18879,2.5},{8.37758,2.5}}}

You can see them on a plot:

Show[Plot[y[x] /. sol, {x, 0, 10}], 
 ListPlot[points, PlotStyle -> Orange]] 

Also, the exact value at any point, e.g. $x=1$ can be retrieved by y[1]/.sol

Sometimes I find it easier to use NDSolveValue which is exactly the same, but gives the values instead of rules. Look it up and you might find it useful.

Answer 2

It would be better if you had supplied an example problem. It is possible that the answer below will not work in general.

Let's use a simple gravity example where we start with an object 10 feet above the ground and let it free fall.

sol = NDSolve[{h''[t] == -98/10, h'[0] == 0, h[0] == 10}, 
  h, {t, 0, 10/7}]

{{h -> InterpolatingFunction[{{0., 1.42857}}, <>]}}

The method I use to define a function of time from this output (there is more than one method) is:

f[t_] := sol[[1, 1, 2]][t]

Now one can plot it:

Plot[f[t], {t, 0, 10/7}]

If we wanted to know at what time the object would be 4 feet above the ground one could use:

solN = NSolve[f[t] == 4, t]
(* {{t -> 1.10657}} *)

This will give you a warning message about Inverse functions but that is OK.