TITLE: How to express a 2nd order ODE as 1st order ODE's?
QUESTION [6 upvotes]: Express the 2nd order ODE
$$\begin{align}\mathrm d_t^2 u:=\frac{\mathrm d^2 u}{\mathrm dt^2}&=\sin(u)+\cos(\omega t)\qquad \omega \in \mathbb Z /\{0\} \\u(0)&=a\\\mathrm d_t u(0)&=b\end{align}$$
as a system of 1st order ODEs and verify there exists a global solution by invoking the global existence and uniqueness theorems.

I'm not sure how to express second order ODEs as first order ODEs, any tips?

REPLY [3 votes]: To convert second-order ODE to a first-order system you have to introduce new variables:
$u_1=u$
$u_2=u'_t$
Now we can write following:
$(u_1)'_t=u_2$
$(u_2)'_t=u''_t=\sin(u_1)+\cos(\omega t)$ 
with the initial condition $u_1(0)=a$  , $u_2(0)=b$