Existence of a solution to a Cauchy's problem on an unbounded interval We know $f$ and $f'y$ are both functions continuous on $\mathbb{R^2}$. Then $\frac{\partial f}{\partial y} \leq k(x)$ where $k$ is also continuous. One needs to prove tha a solution for $$\left\{
  \begin{array}{l l}
    y' = f(x,y)\\
    y(x_0) = y_0
  \end{array} \right.$$ exists for $x_0 \leq x < \infty$
I have absolutely no idea what to do. I look at the Peano theorem, but this just doesn't  apply here, not explicitly at least. Then, I'm not really sure why are the assumptions helpful, especially the one with the partial derivative. Any hints?
 A: You can use an existence theorem with a quantitative estimate for the existence interval. Such a theorem is often called the Picard–Lindelöf theorem but is also associated with other names. In its optimized form it ensures the existence of solution in the interval $[x_0-a,x_0+a]$ provided that $f$ is locally Lipschitz with respect to $y$ (you have that), and there exists $b$ such that $a\le b/M$ where $$M=\max \{|f(x,y)|: |x-x_0|\le a,\  |y-y_0|\le b\}$$
This is a bit complicated to arrange because $M$ depends on $b$. We need some control on $f$ to make this happen. Observe that your assumptions imply
$$
|f(x,y)|\le |f(x,0)|+|y|\, k(x) \tag{1}
$$ 
Fix large $A$; on the interval $[x_0,x_0+A]$ the estimate $(1)$ says
$$
|f(x,y)|\le C_1+C_2 |y| 
$$ 
Therefore, around the point $(x_0,y_0)$ we can have an existence interval of size $$a=(C_1+C_2(|y_0|+1))^{-1}\quad \text{ with } b=1$$ Within this interval $|y|$ is bounded by $|y_0|+1$. Therefore, the next interval will have size $(C_1+C_2(|y_0|+2))^{-1}$, and so on... since the harmonic series diverges, the solution exists on all of $[x_0,x_0+A]$. And since $A$ was arbitrary, this proves the claim.
