I'm struggling in solving this problem. We are given the differential equation $\displaystyle y''=\dfrac{y'}{2\sqrt{y}}$.
We are asked to prove:
a) Any nonconstant solution is strictly monotone.
b) Let's consider the Cauchy Problem
$\begin{cases}y''&=&\dfrac{y'}{2\sqrt y}\\[6pt] y(0)&=&u\\[6pt] y'(0)&=&v.\end{cases}$
Find all points $(u,v)\in\mathbb R_+\times \mathbb R$ such that the maximal solution is nonconstant and defined on the whole real line. For such points $(u,v)$ compute the limits of the solution as $t\to\pm\infty$.
Thanks in advance to anybody who will reply.
-Mario-