Division in differential equations when the dividing function is equal to $0$ When dividing two functions:
$$h(x)=\frac{f(x)}{g(x)},$$
how do we account for the points at which $g(x)=0$ ?
An example is when solving a PDE by separation of variables: Let $\phi(x,y,z)=X(x)Y(y)Z(z)$, then:$$\nabla^2\phi=0\leftrightarrow YZX''+ZXY''+XYZ''=0$$All $\bf {math}$ textbooks, at this step, divide both sides by XYZ, leading to:$$\frac{X''}{X}+\frac{Y''}{Y}+\frac{Z''}{Z}=0$$
But none of them explains why such an operation is valid when there is no requirement for $X,Y,Z$ not to be zeros.
Another exapmple is Sturm-Liouville equation. In some textbooks, it has the form:$$[p(x)y']'+[q(x)+\lambda r(x)]y=0$$
However, other textbooks divide both sides by $r(x)$ and rearrange the terms to obtain another form (without any justification for the division):$$\frac{1}{r(x)}[(p(x)y')'+q(x)y]+\lambda y=0$$ But we all know that a Sturm-Liouville problem can be singular, which means $r(x)$ could be zero at an endpoint. Obviously, the two forms of Sturm-Liouville equation above are not algebraically equivalent to each other.

Most of the time, this is implicit in physics. We just solve the equation without taking into account those issues, and see if it works at the vanishing points.
However, in mathematics, you would need to cut your x interval at the vanishing points, solve in these cut intervals separately, and then study the behavior of the solution at the cuts to see if the problem admits a general solution over all the possible values of x.
