"The" solution is not entirely correct. Every ODE has an infinite family of solutions, usually parameterized by a constant of integration:

is a more general solution. Even works; i.e., the function satisfies the differential equation from the book.

To try to understand the domain where the solution makes sense intuitively, try thinking of solutions as curves (i.e.,integralcurves) in the -plane, and think about the uniqueness of solutions. Given any point where , you can can find a constant such that the curve

passes through the point . However what happens at ? For any (other) constant , the curve , , is also a solution to the DE, and it meets the solution at the origin. Uniqueness is violated.

For any point in the domain , there is a unique solution of the DE passing through that point that is defined for all . The same is true for the domain . This is why the authors exclude .

Hope that helps!

Jerry