Discontinuous solutions of a linear ode with singularities I've came across the following ode in my studies
$$a(t)\dot{x}(t) = A(t)x(t),$$
where $x\in \mathbb{R}^m$, $A(t),a(t)$ are analytic and $a(t) \geq 0$. 
I would like to understand what happens to a solution when it passes through a moment $t_0$ when $a(t_0) = 0$. In particular for other reasons it seems that there might be a weak solution with a jump discontinuity, but I am not able to verify this directly.
So I was wondering, if maybe someone stumbled upon this type of ode's and could give me some references about known results for this type of equations. Or maybe some advise on how one could study it's solutions.
Thank you.
 A: There are basically two situations. 
Case 1: $A(t_0) \neq 0$. 
In this situation by assumption $a(t) \geq 0$ we have that $a(t)$ is approximately $\frac12 a''(t_0)(t - t_0)^2$ near $t_0$. So to model the situation we can look at the equation 
$$ t^2 \dot{x} = x $$
This we can solve to be 
$$ x(t) = C \exp(- 1/t ) $$
on each regime $t\in (-\infty,0)$ or $t\in (0,\infty)$ separately. In particular, if on the $t < 0$ regime we choose $C$ to be non-zero, then $x$ must blow-up at $t = 0$. However, if you choose $C = 0$ on the $t < 0$ regime, you can get an infinite family of smooth solutions to this ODE, parametrized by $C_+\in \mathbb{R}$, given as
$$ x(t) = \begin{cases}
0 & t \leq 0\\
C_+ \exp(-1/t) & t > 0 
\end{cases} $$
A similar analysis holds for more general $A$ and $a$: near $t_0$ if $A(t_0) \neq 0$, then $A / a$ is not integrable. Then one of the left/right limits must blow-up unless to the left/right the solution vanishes identically; on the other side however we have non-uniqueness, in that we get a one parameter family of solutions that smoothly continues past the singular point. 
Case 2: $A(t_0) = 0$. 
In the case where $A/a$ has a removable singularity at $t_0$, the equation is essentially regular and there exists a unique $C^1$ solution to the ODE for each value of $x(t_0)$. But one can solve the ODE to the left and right of $t_0$ with different initial conditions and still satisfy the equation $a\cdot{x} = Ax$. This gives possible discontinuous solutions. The simplest example is 
$$ t^2 \cdot{x} = t^2 x $$
The $C^1$ solution is $x(t) = x(0) \exp t$. But formally you can consider weak solutions with $x(t) = x_- \exp t$ for $t < 0$ and $x(t) = x_+ \exp t$ for $t > 0$. 
Another regime is the borderline case where $A / a$ is singular at $t_0$, but the singularity is borderline non-integable. More precisely, we are looking at the situation that $A / a$ has a simple pole at $t_0$. In this case the model equation is
$$ t^2 \dot{x} = t x $$
which we can solve to get
$$ x = Ct $$
for arbitrary constant $C$. In this situation necessarily any solution must converge to $0$ at $t \to t_0$; again we have a unique $C^1$ solution to the differential equation, but we also have nonuniqueness of a different type: here, instead of $x$ being discontinuous, solutions are all continuous, but its first derivative may be discontinuous at the singular point. 

In terms of techniques: separation of variables 
$$ \frac{\dot{x}}{x} = \frac{A}{a} $$
works pretty well away from the singular points; this gives you description of the dynamics as you approach the singular points, and taking the limit you can analyze the limiting behaviors and how solutions to the left and right of the singular points can be joined together to get various types of weak solutions. 
