Consider the nonlinear ODE $$y\cdot\frac{d^{2}y}{dx^{2}}-\left(\frac{dy}{dx}\right)^{2}+1=0$$ with $y : \mathbb{R} \to \mathbb{R}$.
As seen, it does not satisfy Picard–Lindelöf theorem because of singularity at $y=0$. Nevertheless it possesses smooth solutions, for example $y(x) = \frac{1}{a}\cos ax$ for $a \ne 0$. On the other hand, all the solutions of $$y\cdot\frac{d^{2}y}{dx^{2}} + \left(\frac{dy}{dx}\right)^{2}+1=0$$ explode when $y$ approaches $0$.
The first singularity leads to violation of uniqueness, while the second breaks the solutions.
Are there special terms for these types of singularities? A study? The term "regular singular point" seems to have a different meaning.
I expect a term for the singularity of the first equation like "passable" or "permeable", but I didn't find any of them in math literature.
That might be a coordinate or removable singularity. With some change of variables, there seems to be a meaningful value at $y=0$. Either way I think its solidly a Type I singularity: Mathematical Singularities
$y\frac{d^2y}{dx^2}-(\frac{dy}{dx})^2+1=0$
$\frac{d}{dx}(\frac{dy/dx}{y})=-1/y^2$
$=\frac{d^2}{dx^2}(\ln y)=-1/y^2$
$u=\ln {y}$
$\frac{d^2}{dx^2}u=-e^{-2u}$
$\frac{du}{dx}\frac{d^2}{dx^2}u=-e^{-2u}\frac{du}{dx}$
$(du/dx)^2=c_1+e^{-2u}$
$\frac{du}{\sqrt{c_1+e^{-2u}}}=dx=\frac{dy}{y \sqrt{c_1+1/y^2}}=\frac{dy}{\sqrt{c_1y^2+1}}$
$\sqrt{c_1}y=\tan\theta\implies dy=\sec^2{\theta}d\theta/\sqrt{c_1}$
$\frac{1}{\sqrt{c_1y^2+1}}=\cos{\theta}$
$\int dy/[...]=\int(1/\sqrt{c_1})\sec{\theta}d\theta=\frac{1}{\sqrt{c_1}}\ln|\frac{1+\sqrt{c_1}y}{\sqrt{1+c_1y^2}}|+c_2=x$
Plug $y=0$ into that, you get $x=c_2$.
$\frac{1+\sqrt{c_1}y}{\sqrt{1+c_1y^2}}=c_3e^{\sqrt{c_1}x}$
$[(1+c_1y^2)c_3^2e^{2\sqrt{c_1}x}-1]^2=c_1y^2$
This will give you a fourth order polynomial in y with coefficients consisting of an exponential function of x.