Optimizing a function involving NDSolve results in error messages

Question

I want to minimize a function f with NMinimize but an error occurs that says:

f[l_] := (
  kk = NDSolve[{y'[x] == 2*x + 3, y[l] == 0}, y, {x, 0, l}];
  out = Evaluate[y[0] /. kk];
  out[[1]])

NMinimize[{f[t], 0 <= t <= 10}, f, t]

During evaluation of In[18]:= NDSolve::ndsv: Cannot find starting value for the variable y. >>

During evaluation of In[18]:= ReplaceAll::reps: {NDSolve[{(y^[Prime])[x]==3+2 x,y[{t}]==0},y,{x,0,{t}}]} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. >>

During evaluation of In[18]:= NMinimize::argrx: NMinimize called with 3 arguments; 2 arguments are expected. >>

Out[18]= NMinimize[{y[0], 0 <= t <= 10}, f, t]

How can I fix the function f to avoid this problem?

Answer 1

Check this

f[l_?NumericQ] := Module[
{kk, y, x, out},
kk = NDSolve[{y'[x] == 2*x + 3, y[l] == 0}, y, {x, 0, l}];
out = Evaluate[y[0] /. kk];
out[[1]]
];
res=NMinimize[{f[t], 0 <= t <= 10}, t]

it gives

{-130., {t -> 10.}}

You can check if the answer is correct by plotting your function.

Show[Plot[f[t], {t, 0, 10}, Frame -> True], 
Graphics[{Red, PointSize[Large], Point[{res[[2, 1, 2]], res[[1]]}]}]]

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