Differential equations: Shouldn't the method of separation of variables miss the solutions when x=0? This question is regarding "Example 6" from the book "Elementary differential equations with boundary value problems" by Edwards and Penney.
Consider the ODE:
$x \frac{d y}{d x}=2 y$    (1)
We apply the separation of variables technique to obtain:
$\frac{1}{y}dy=\frac{2}{x}dx$ (2)
which provides the general solution:
$y(x)=C x^{2}$  (3)
The book mentions that "the general solution (3) satisfies (1) for any value of the constant C and for all values of the variable x".
However, I believe the statement is wrong when $x=0$. By using the separation of variables, we divided by $x$ and $y$ to obtain (2). Thus, we make the implicit assumption that $x \neq 0$, and the separation of variables will generate solutions where $x \neq 0$.
How can $y$ be defined for any $x$, when we implicitly assume that $x \neq 0$?

You are correct, as an ordinary DE the domain does not contain the vertical line $x=0$. It is a special feature of this equation that you can take two solutions $=C_1x^2$ for $x<0$ and $=C_2x^2$ for $x>0$ and connect them to a differentiable function at $x=0$. But there is no innate connection between the two constants, one can take any random combination.
