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I have the following differential equation:

a*L'[t] == 40000*L[t]^(-3/2) - 1.4 - extf

I want to fit the solution to data in order to acquire a. I tried DSolve with a=1 and separation of variables:

Clear[t]
Clear[L]
extf = 15
tmax = 100
solt = Integrate[1/(40000*L0^(-3/2) - 1.4 - extf), {L0, 10, L}, Assumptions -> 10 < L < 181.194];
solL = L -> InverseFunction[Evaluate[solt] & /. L -> #]
N[L[1000] /. solL]

 OUTPUT=  -16618.8 - 341.659i

This is wrong (should be around 170..) so that I tried NDSolve:

Clear[t]
Clear[L]
Clear[a]
extf = 15
tmax = 100
solution = ParametricNDSolve[{a*L'[t] == 40000*L[t]^(-3/2) - 1.4 - extf, 
           L[0] == 10}, L, {t, 0, tmax}, {a}]

Manipulate[ Plot[Evaluate[L[a][t] /. solution], {t, 0, tmax}, 
                  PlotRange -> All], {a, 1, 2}]

That gives me the right graph but I cannot fit this result to my data to obtain the right a nor is it as nice as a analytic expression.

What is going wrong with DSolve?

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DSolve can solve a generalized equation:

sol = DSolve[{a*L'[t] == b*L[t]^(-3/2) - c, L[0] == L0}, L, t]
(*
  {{L -> Function[{t}, InverseFunction[... &][-(t/a) +...]]}}
*)

Block[{b = 40000, c = 1.4 + extf, L0 = 10},
 Plot[Evaluate[L[t] /. sol /. {{a -> 1}, {a -> 2}}], {t, 0, tmax}, PlotRange -> All]
 ]

Mathematica graphics

It agrees with the OP's numeric solution:

Plot[Evaluate[{L[1][t], L[2][t]} /. solution], {t, 0, tmax}, PlotRange -> All]

Mathematica graphics

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