Here is the ode I'm interested in:
$\frac{ \partial y}{\partial x}= (1+y)^{\frac{3}{4}}(1-y)^{\frac{3}{4}}\;$
I would like to know, in which way and when, the solution of the above ode approaches $\;\pm 1$.
My attempt:
Since the solution cannot be expressed in closed form (the solution according to Wolfram Alpha is given by : $c+x=y(x)_2F_1(\frac{1}{2},\frac{3}{4};\frac{3}{2};{y(x)}^2)\;)$, my professor suggested to compare the behavior of the above solution when $y\sim 1\;$ with the solution of the following I.V.P:
$\frac{ \partial q}{\partial s}=4q^{3/4}\;\\q(0)=0\;$
by setting $q=1-y\;$
I have no idea how should I proceed. I solved the problem (which hasn't a unique solution because the Lipschitz condition isn't satisfied) but I have no clue how to compare this solution to the first one, in order to find more information.
I would really appreciate if somebody could help me through this. How should I handle similar to this, problems? Is there something I should check or calculate first?
I'm totally lost, so any help would be valuable.
Thanks in advance