# Function output from DSolve

I want to get a function as output form DSolve. For Example :

sol = DSolve[{Q''[t] + 40 Q'[t] + 625 Q[t] == 100*Cos[10*t], Q[0] == 0, Q'[0] == 0}, Q[t], t]


I want to use Q[t] as a function to use in an another equation.

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–  Mr.Wizard Jul 24 '12 at 13:32
Almost the same question: How to use results of NDsolve for further solving of ODEs? –  Jens Jul 24 '12 at 17:40

It's not too hard:

dsaad[t_] = Q[t] /. First @ DSolve[{Q''[t] + 40 Q'[t] + 625 Q[t] == 100*Cos[10*t],
Q[0] == 0, Q'[0] == 0}, Q, t]


Notes:

1. I instructed DSolve[] to return the pure function Q (through the second argument) instead of the function itself to ease the replacement.

2. I used Set[] (=) instead of SetDelayed[] (:=), so that the replacement is done at once before the definition takes place.

Another possibility is to use DifferentialRoot[] instead:

dsaad[x_] := DifferentialRoot[Function[{Q, t},
{Q''[t] + 40 Q'[t] + 625 Q[t] == 100*Cos[10*t], Q[0] == 0, Q'[0] == 0}]][x]

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I understand all of it but I didn't get the "First". What does it? –  DSaad Jul 24 '12 at 14:28
@DSaad The First is meant to extract the solution as it is returned by Mathematica as a List of List. –  PlatoManiac Jul 24 '12 at 15:17