Is Lipschitz condition necessary for existence and uniqueness of a solution for a first order differential equation? Is the Lipschitz condition only a sufficient condition for existence and uniqueness of $\frac{dy}{dx}=f(x,y), y(0)=y_0$? Is it also necessary?

Take the case of separable equations:
$$\frac{dy}{dx} = a(x) b(y)$$
with $x(x_0) = y_0$. Asume that $a>0$ around $x_0$. The equation is equivalent to
$$\int_{x_0}^x \frac{1}{a(t)} d t = \int_{y_0}^y b(t) dt$$
as a functional relationship between $x$ and $y$. For the existence and uniqueness of this type of Caychy problem near $(x_0, y_0)$ it is enough that $a>0$ and continuous, and $b$ is continuous. So Lipschitz in $y$ is not required. 
