Function output from DSolve

Question

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.

Answer 1

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.

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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]