In my Quantum Mechanics textbook, they present the following differential equation
$$dx = (\frac{y}{a-y})^{0.5} dy$$
and offer solutions to the system, including a cycloid (which was easy to verify), as well as the curve $y(x)=0$. But I do not understand why they consider $y(x)=0$ to be a solution. It should be obvious that the right side of the equation would be $0$, and that the left side should be $1$, right? What am I missing? The mathematics are just glossed over in this example.
Thank you,
XeB
Edit: Forgot to specify, $a \neq 0$.
Your analysis is correct, $y\equiv 0$ is not a solution. It is even more apparent if you write the equation with derivatives instead of differentials $$y'(x) \sqrt{\frac{y}{a - y}} = 1,$$ then, if you plug $y\equiv 0$, you obtain the contradiction $$0 = 1.$$