Existence of a solution for a nonlinear ODE on $[0,\infty)$ I'd like to prove that the solution to the following IVP exists on $[0,\infty)$. The IVP is given by
$$ \begin{cases} 
      y'(t) = y^2 \cos(t)-ye^t  \\
      y(0)= y_0
   \end{cases}
$$
where $y_0 \in \mathbb{R}$.
I've already established unique solvability in an interval around the origin, say $[0,\epsilon)$ for some $\epsilon>0$ because $f(y,t)=y^2 \cos(t)-ye^t$ is lipschitz in $y$ in a neighborhood of the origin.
My usual strategy for showing a solution exists on such an interval is to try to find a upper/lower solution to make a bound for the solutions and thus use the bound to generate some information about what happens to the solution as $t \rightarrow \infty$. This, however, is difficult in this case because of the $e^t$ term.
How might I show existence on $[0,\infty)$ given any initial condition, for $y_0$?
 A: It's not true.  The general solution to your differential equation is
$$y(t) = \dfrac{-\exp(-e^t)}{\int \exp(-e^t)\cos(t)\; dt}$$
where the denominator is any antiderivative of $\exp(-e^t)\cos(t)\; dt$.
In particular, taking an antiderivative that is $0$ at, say, $t=1$, you get
a solution that becomes infinite at that value of $t$.  Numerically, this corresponds to the solution with initial value 
$$y(0) = \dfrac{e^{-1}}{\int_0^1 \exp(-e^s)\cos(s)\; ds} \approx 2.037005842$$ 
A: HINT: You can use Peano theorem to prove (local) existence of solution for your ODE:

If $D\in \mathbb{R} \times \mathbb{R}$  is an open subset of $\mathbb{R}^2$, and if $f:D \to \mathbb{R}$ is continuous, then the ODE 
  $$
\begin{cases}
y'(t) = f\big(t, y(t) \big), \\
y\left(t_0\right) = y_0, & \left(t_0, y_0\right) \in D,
\end{cases}
$$
  has a local solution $\hat y : [t_0-\varepsilon, t_0 + \varepsilon]  \to \mathbb R$ for some $\varepsilon > 0$.

You will probably have to check that your ODE satisfies statement of the theorem, which should not be difficult at all. 
Alternatively, you can look up proof of Peano theorem and use it to construct custom proof for your particular ODE.

PS As pointed out in the comments by @RobertIsrael, this approach only allows to establish existence of local solution only. 
Global solution, however, may not exist.
