Why are we allowed to divide by $x$ in the following differential equation?

Question

Suppose we have a second order linear differential equation $x\frac{d^2y}{dx^2}+\frac{dy}{dx}-xy=xe^x$.Often we divide both sides by $x$ to get $\frac{d^2y}{dx^2}+\frac{1}{x}\frac{dy}{dx}-y=e^x$,and then proceed towards solution.Why does this make no harm as far as the solution of the differential equation is considered.We were trying to find $y=\phi(x)$ whose domain is $\mathbb R$.But,now we are finding a solution $y=\psi(x)$ whose domain is $\mathbb R-\{0\}$.Of course it is clear that the family of solutions $\phi(x)$ and $\psi(x)$ are equal except one has $0$ in domain and other does not ,I mean we can extend the other one to $\mathbb R$ by assigning some value at $0$ so that the extension is continuous.Does this make sense?Is this why we can divide by $0$ without worrying about losing anything?

Answer 1

Of course you have to worry about dividing by $x=0$. Having $x$ as a coefficient (especially of the highest-order term) is often a sign that you should expect some kind of weird behaviour of the solutions at $x=0$.