these might sound like extremely trivial questions but since my background is more in probability and statistics I'm not too sure what to do, or even what to read up on to understand what to do. I believe that my equation might be behaving irregularly due to singularity point(s), however, I don't know how to test the equation to discover whether there are singularity points in it.
By singularity I mean as defined in the article: https://en.wikipedia.org/wiki/Singularity_(mathematics)
Say for instance I have an equation of the form:
$$f(x,0) = a( b(x) - c(x))$$
where $a$ is a constant, and $b(x)$ and $c(x)$ are functions (to be precise, $b(x)$ and $c(x)$ are both very simple first order PDEs). Now, for this equation, whenever I put particular values into $b(x)$ and $c(x)$, then the equation $f(x,0)$ will not have any singularities, but for certain other values which are put into $b(x)$ and $c(x)$ then there are singularities. For example, say for instance I put $x^{2}$ into the functions $b(x)$ and $c(x)$, then the function $f(x,0)$ will not have any singularities. However, if I instead put $\log(x)$ into $b(x)$ and $c(x)$ then the function $f(x,0)$ will now have a singularity.
Now, my questions are:
Firstly, is there any way for me to test the equation to determine whether there are any singularities in it? Is there a theorem or a method or something which I can apply to the equation which will allow me to check whether there are any singularities in it? In particular, since $b(x)$ and $c(x)$ are a type of PDE, is there a theorem which says something like "For a PDE to have a singularity it must satisfy the following properties:... " or something like that? (I know that's not realistic but I just meant that as an example).
Secondly, if I do find singularities in the equation is there any way for me to remove them?
Thanks in advance.