lteqn =
Not sure the underlying reason behind it but you are defining the X as a function of s and solve function seems to be having trouble solving it. Define X as a standalone and it has no trouble
syms t x(t) s X;
% PARAMETERS (tried to do symbolically but it was more diffcult)
m = 1;
k = 0.5;
xi = 1.2;
c = xi*2*sqrt(k*m)
f0 = 1;
w = 0.1;
dx = diff(x, t, 1);
ddx = diff(x, t, 2);
% INITIAL CONDITIONS
x0 = 0;
dx0 = 0;
newton = m*ddx+ c*dx +k*x;
f = f0*cos(w*t);
lteqn = laplace(newton, t, s);
lefteqn = subs(lteqn,{laplace(x(t), t, s), x(0),dx(0)},{X, x0, dx0});
F_s = laplace(f, t, s);
simplify(solve(lefteqn == F_s, X))
