The RHS of your differential equation should be 3*u.
In the frequency domain, Y(s) = H(s)*U(s). It looks like you're trying to get the step repsonse of your second order system, so U(s) = 1/s. You forgot to multiply by U(s) in your symbolic approach.
num = 3;
den = [1 2 5];
nums = poly2sym(num,s);
dens = poly2sym(conv(den,[1 0]),s);
3/5 - (3*exp(-t)*(cos(2*t) + sin(2*t)/2))/5