Suppose that we have a singular BVP of the form $$\dot{x}=\frac{f(x,t)}{t^n}, \qquad x(0)=x_0$$ where $x\in \mathbb{R}^n$, $n\in \mathbb{N}$.

There is of course no more uniqueness and the existence is not obvious as well. But what I would like to do is to consider instead a regularized version of the problem: $$\dot{x}=\frac{f(x,t+\varepsilon)}{(t+\varepsilon)^n}, \qquad x(0)=x_0$$ and to give enough information to describe the pointwise limit $\lim_{\varepsilon \to 0+}x(\varepsilon,t)$.

For example, I have found that for a one dimensional BVP problem of the form $$ \dot{x} = c - \frac{x^2}{(t+\varepsilon)^2}, \qquad x(0)=x_0, $$ $x(\varepsilon,t)$ will converge pointwise for $t>0$ to a unique solution of the above equation with $\varepsilon =0$ and boundary conditions $x(0) = 0$, $\dot{x}(0) = -(1+\sqrt{1+4c})/2$. The problem is that I was able to do this one as a very-very special case. For example, I am completely stuck if we add two extra equations like $$ \dot{x} = c - \frac{x^2}{(t+\varepsilon)^2}, $$ $$ \dot{y} = \frac{xy}{(t+\varepsilon)^2}, $$ $$ \dot{z} = d - \frac{y^2}{(t+\varepsilon)^2}. $$

**So my question is:** has anyone seen some similar problems and techniques to solve them in the literature? Any ideas or references would be very helpful.