I have always wondered about the precise terminology when it comes down to differential equations. When one talks about a solution curve does this imply that the curve is a function that is well defined (in terms of each $x$ having a maximum of one value of $y$ to map to)?
For example, if we have the differential equation
$\frac{dy}{dx}2y=1$ the solution is
$y^2=x+c$. Are the solution curves (three branches?) that are $y=\sqrt (x+c)$ and $y=-\sqrt(x+c)$ for $x>-c$ corresponding to different initial conditions ?
Thank you in advane.
