Why does the general solution of $y'=y$ not covering $y=0?$ It is regarding the ODE $y'=y.$ 
Usually we try to find its general solution using separation of variables as 
$\frac{dy}{y}=dx\implies\log y=x+c\implies y=e^{x+c}$ ($c$ being arbitrary constants).
Please tell me why does the general solution does not cover the case $y=0.$

Generally, when solving ODEs, you are looking for a non-trivial solution; i.e. $y$ that is not identically zero. Note that the trivial solution ($ y = 0$) is quite often a valid solution, but it just isn't interesting from either a mathematical or a practical point of view. Moreover, as others have mentioned, the method breaks down if we do not assume that $y$ is not zero.
