I am interested in learning whether there is a standard, systematic way of determining the singular points of non-linear, second degree ODEs.
Particularly, I am interested in determining at what time this autonomous equation will blow-up, without actually having to solve the equation: $$ x''(t)=\frac{(x'(t))^2}{3}+e^{x(t)}\quad;\quad x(0)=0,\quad x'(0)=0~~.\tag{1}$$ Equation (1) is actually solvable by very clever substitutions (see here, in eqworld), but I am interested more in the theory and more flexible procedures -- or useful theorems.
I know Equation (1) will blow-up, because the solution to the related equation $$\chi''=e^\chi\quad;\quad \chi(0)=0,\quad \chi'(0)=0~~.\tag{2}$$ diverges at $t=\pi/\sqrt{2}$. Right side of Equation (1) grows even faster than (1), so it should blow-up at $t<\pi/\sqrt{2}$ as well, as numerical exploration indicates. We can prove that Equation (2) blows-up by noting that $$ (\chi')^2=2(e^\chi-1)~~.$$ Then, we calculate the integral $$\int_0^\infty \frac{dz}{\sqrt{2(e^z-1)}}=\frac{\pi}{\sqrt 2}~~,$$ from where we can deduce that solution to Eq. (2) blows-up at $\pi/\sqrt{2}$ without actually having to solve (2). Can we maybe adapt something like this to second order?
It goes without saying: any help, comment, question, or thought will be very appreciated.
A general theory of finite-time singularities for dynamical systems seems to have been in active development in the late 90's. However, most of what is known about the issue is in terms of polynomial systems, i.e. $\dot{x} = P(x)$ for $x\in\mathbb{R}^n$ and $P$ a polynomial. For transcendental functions such as what you have here, your best bet is to find an analytic solution whenever possible.
If you're interested in seeing the literature for polynomial systems, you can check out these two papers:
Necessary and Sufficient Conditions for Finite Time Singularities in Ordinary Differential Equations (1998)
Finite time blow-up in dynamical systems (1999)
The second paper presents a method of evaluating the time of blow-up, though I can't speak for its generality. I hope you get some use out of this.