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))