Hi Tim,

the reason is that variables controlled by Manipulate are local within Manipulate. To see this try:

f[x_] := Sin[a x]
Manipulate[a f[x], {x, 0, Pi}, {a, 1, 2}]

Consequently the 'a' inside Sin and 'a' inside Manipulate are different variables.

Cheers Henrik

In the first part of the code you can replace the alphas, etc, by adding a 0 to the end:

Eqn1 = D[f[t], {t, 2}] + \[Omega]0^2*f[t] == 0;

Sol1 = DSolve[Eqn1, f[t], t];

IC1 = f[0] == \[Alpha]0;

IC2 = (D[f[t], {t, 1}] == \[Gamma]0) /. t -> 0;

Sol2[t_] = Simplify[DSolve[{Eqn1, IC1, IC2}, f[t], t]];

x[t_] = Replace[f[t], Sol2[t]];

Then this works:

Manipulate[{Plot[
   x[t] /. {\[Omega]0 -> \[Omega], \[Gamma]0 -> \[Gamma], \[Alpha]0 \
->  \[Alpha]} , {t, 0, 10}], 
  x[1] /. {\[Omega]0 -> \[Omega], \[Gamma]0 -> \[Gamma], \[Alpha]0 -> \
 \[Alpha]}}, {\[Omega], 1, 5, 1}, {\[Gamma], 1, 5, 1}, {\[Alpha], 1, 
  5, 1}]

Should help you figure what was wrong.