clear
reset(symengine)

syms t real
syms w positive

M = [ 
   0   1 ; 
  -w^2 0 
]


[U,D] = eig(M)

B = U * expm(D*t)

syms x0 dx0 real

q0 = [x0 ; dx0]

alpha = inv(U) * q0

q_general = simplify( B * alpha )