Retain the zero solution when separating an exponential equation

For $y\ne0$, $dy/dx=2y$ gives $\int(1/y)\,dy=\int2\,dx$, so $\ln|y|=2x+A$. Exponentiating gives $|y|=e^{2x+A}$, hence $y=\pm e^Ae^{2x}=Be^{2x}$ with nonzero real B. The sign and exponential constant can be absorbed into one new constant.

The constant function $y=0$ also satisfies $y\prime-2y=0$. Combining both cases, $y=Ce^{2x}$ for every real C. The source calls the zero case “trivial” and remarks that examinations discuss nontrivial cases; that remark does not justify dropping a valid solution from this unrestricted general-solution ask.