Solve the equation $a(x)y' + b(x)y=c(x)$. What if $a(x)=0$ somewhere? Does it always blow up? 
Solve the equation $$a(x)y' + b(x)y=c(x)$$ (Hint: integrating factor.)

I believe that if we assume $a(x)$ is differentiable and nonzero on an interval $(\alpha, \beta)$, we can use the integrating factor $$p(x):= \exp\left[\int_\kappa^x \frac{b(\xi)-a'(\xi)}{a(\xi)} d\xi\right]\quad \text{for any }\kappa \in (\alpha, \beta)$$ and arrive at the solution $$y=\frac{\int_\kappa^x p(\eta)c(\eta)d\eta}{p(x)a(x)}+\frac{\omega}{p(x)a(x)}\quad \text{ for } \omega\in (\alpha, \beta).$$
I am wondering how to treat the case where $a(x)$ may be zero (let's assume $a(x)$ is differentiable on $\mathbb{R}$). I observe that we cannot expect there to be a continuous solution in general on $\mathbb{R}$ (Edit: or will it? $\sqrt{x}$ has unbounded derivative near zero, but does not blow up). A counterexample is if $b(x) \equiv 0$, $a(\rho)=0$, $c(\rho)\neq 0$, then the solution will blow up at $\rho$. But what if $b(\rho) \neq 0$?
Since I'm sure this is a common topic, a reading reference would be fine in lieu of an answer.
 A: One thing to observe is that we cannot solve a general initial value problem at a point $x_0$ with $a(x_0)=0$, as the value of $y(x_0)$ is determined by the equation. The question is whether we can have any continuous solution at all. 
Not in general. Consider $$xy'+by=0\tag{1}$$ with constant $b\ne 0$. Then $(\ln y)'=y'/y = -b/x$, hence $y = Cx^{-b}$. Solutions will blow up at $0$ if $b>0$, and will be continuous if $b<0$. (For $x<0$ one should interpret $x^{-b}$ as $\pm |x|^{-b}$; the choice of sign is yours.)
Making the right-hand side nonzero won't change anything: e.g., $xy'+by=1$ reduces to the above by substitution $z=y-1$. 
Such equations are usually studied when the coefficients are meromorphic functions: then one can divide by $a$ and still have meromorphic coefficients, then look for a power series solution by solving for coefficients recursively. This works better when the singular point is regular; the Wikipedia article has a few references. Of those, I looked at Chapter 4 of Teschl's book; it's quite easy to read. I recommend this book for ODE in general.  
