# How to combine DSolve and FindRoot

As my differential function is relatively complex I do not want to do the detour of finding an explicit solution via DSolve for it but rather use an FindRoot-like approach by guessing initial starting values.

Following example:

Let z[t] be the differential where the start and end values are known:

z[0]=z0 and z[tmax]=zend, but the end time tmax is unknown.

The (simplified) differential (where the paths of l2[t]and l3[t] as well as their end values l2final and l3final are well-defined) is:

DSolve[{z[0] == z0, z[tmax] == zend, z'[t]==f[l20[t],l3[t]],l2[tmax]==l2final, l30[tmax]==l3final}, z[t],t]


The only unkown is tmax. So I don't want to do the DSolve described as I don't care about an explicit solution of z[t]. But I rather want to search for something like:

FindRoot[z[t]-zend,{t,0,100}]


Does anybody have an idea how to combine both approaches EFFICIENTLY?

Btw: z[t] is continously increasing as well as l2[t] and l3[t]

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It helps to give code that can reproduce you're problem, but here's a guess at what you're after. I removed the conditions l2[tmax]==l2final, l30[tmax]==l3final because if l2 and l3 are well defined and tmax is defined by z[tmax] == tend, then l2final and l3final are determined by the result.

WhenEvent is the way to solve z[t] == zend.

f[x_, y_] := x + y^2/4;
l20[t_] := t + Cos[4 t]^2; l3[t_] := 2 t + Sin[2 t];
z0 = 10.;
zend = 100.;
sol = First @ NDSolve[{z[0] == z0, z'[t] == f[l20[t], l3[t]]
WhenEvent[z[t] == zend, tmax = t; "StopIntegration"]},
z[t], {t, 0, Infinity}]

tmax

(*
InterpolatingFunction[{{0.,1.76425}},<>]
5.97248
*)

Plot[sol[t], Evaluate[{t, Sequence @@ First@sol["Domain"]}]]


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