Explain-global existence and uniqueness theorem

Question

Explain the steps on how to solve the problem below:

Express the 2nd order ODE

d_t² u=d² u/dt² =sin?(u)+cos?(ωt) ω ∈ Z/{0}

u(0)=a

d_tu(0)=b

as a system of 1st order ODEs and verify that there exists a global solution by invoking the global existence and uniqueness Theorem.

Useful information:

Global existence and uniqueness Theorem:

The ordinary differential equation

d_tu_–=f_–(t,u_– (t))

u_–(0)=u_–0

has a unique solution if f_–∈C⁰ (I)×Lipschitz(L_∞ (R)), f is continuous with respect to 1st variable and Lipschitz with respect to 2nd variable.

Lipschitz Continuity: A function g:I→R is Lipschitz continuous if ∃Λ>0 such that ||g(_x–)-g(y_–)||≤Λ||_x–y_–||∀x_–,_y–∈I.

NB: means vector value.