Q:

How to show that a solution blows up

I am doing this problem. For part a, I think I can just let $y(t)=at^2+bt+c$, but the problem is, how can I make sure what is the max precision of $y(t)$? I am not sure if it is of degree 2 or 3 or 4, but since I am only asked to find $y(0)=s$, I suppose it doesn't matter?
And for part b, can anyone please help? Thanks a lot!

A:

Your solution is wrong. Just by inserting it you will find that a quadratic polynomial is no solution.
"Blowing up in finite time" means that the solution runs into a singularity, most often a pole. A quadratic polynomial does not have a pole.
You can find the exact solution by separation of variables and partial fraction decomposition or by treating the ODE as a Bernoulli equation.
Using $y=u^a$, $y'=au^{a-1}u'$ you get
$$
au^{a-1}u'=\frac12 (u^a+u^{3a})\iff au'=\frac12 (u+u^{2a+1})
$$
which simplifies for $a=-\frac12$ to
$$
u'=-u-1\iff u=-1+Ce^{-t}\implies y(t)=\frac1{\sqrt{Ce^{-t}-1}}
$$
so that $C-1=s^{-2}$ and consequently the singularity appears at $t=\ln(C)=\ln(1+s^{-2})$.

