Finding parameters for first positive root of solution to differential equation to be equal to a desired value

Question

I have a differential equation with parameters. The objective is to vary the parameters such that the first positive root of the solution is equal to a desired value obj.

As the equation is very involved, I decided that I'd use user1084363's question in Find all roots of an interpolating function (solution to a differential equation), which inspired the first part of my solution.

Currently, my solution is to manually manipulate the parameter k to obtain the desired solution. For example, if I'd like the first root of the equation to be 1.6, I would use the following line of code

obj=1.6;
Manipulate[Flatten[Reap[NDSolve[
{1.09 x''[t] - k* x'[t] + 1.1759 Sin[x[t]] == 0,  x[0] == Pi/3, x'[0] == 0}, x, {t, 0, 50}, 
Method -> {"EventLocator", "Event" -> x[t],"EventAction" :> Sow[t]}]]][[2]]-obj, 
{k, 0.01, 0.1}]

and manually vary the value of k till I obtain a value of zero.

However, this is quite time consuming and I was wondering whether there was a way to automate the process. My first thought was to use Root, with k as the variable, but this doesn't work as of now due to the presence of the differential equation.

Answer 1

I'd do :

auxSol = ParametricNDSolve[{1.09 x''[t] - k  x'[t] + 1.1759 Sin[x[t]] == 0, 
                             x[0] == Pi/3, x'[0] == 0}, x, {t, 0, 50}, {k}]

data = {#, FindRoot[(x[k][#] /. auxSol) == 0, {k, 0.01, 0.1}][[1, 2]]} & /@ Range[1, 2, 0.05]

ListPlot[data, Filling -> Axis, AxesLabel -> {"obj", "k"}]