Fit to inverse function

Question

I am trying to fit my data to an inverse function from solving a DGL:

extf = 15
data = Import["http://pastebin.com/raw.php?i=mTQUNBua", "Table"];
ListPlot[data]
sol = DSolve[{a*L'[t] == b*L[t]^(-3/2) - c, L[0] == L0}, L, t]
Block[{b = 40000, c = 1.4 + extf, L0 = 10}, 
       fit=FindFit[data, L[t] /. sol, {a}, t]]
Plot[fit,{t,0,100}]

Obviously this takes a huge amount of time. But after a while weird error messages start to show up. Is this in principle the right approach?

Answer 1

data = Import["http://pastebin.com/raw.php?i=mTQUNBua", "Table"];
b = 40000; c = 1.4 + 15; L0 = 10; 
fun = L /. ParametricNDSolve[{a*L'[t] == b*L[t]^(-3/2) - c, L[0] == L0}, L, {t, .2, 2000}, {a}]
fit = FindFit[data[[1 ;; -1;; 10]], {fun[a][t], 9 < a < 15}, {{a, 10}}, t]

(* {a -> 14.026} *)

Show[Plot[fun[a /. fit[[1]]][t], {t, .2, 2000}, PlotRange -> All, PlotStyle -> {Thick, Red}],
    ListPlot[data[[1 ;; -1 ;; 100]]]]