Inconsistency for solving $x' = x^{1/2}$ The proposed system $x' = x^{1/2}$ can be solved easily to obtain $x(t) = \frac{1}{4} (t^2 + t c + c^2)$, where $c$ is the integration constant.
However, differentiate the newly-found $x(t)$, one gets: $x(t)' = \frac{1}{2}t+\frac{1}{4}c$. This implies that $x^{1/2} = \frac{1}{2}t+\frac{1}{4}c$. However, 
$$(x^{1/2})^2 = \left(\frac{1}{2}t+\frac{1}{4}c\right)^2 = \frac{1}{4}\left(t^2 + \frac{1}{2}tc + c^2\right) \neq \frac{1}{4} \left(t^2 + t c + c^2\right) = x(t)$$
Does anyone know why the inconsistency occurs? I understand there is another solution, but it is also inconsistent.

By separation of variables:
$$
\frac{dx}{\sqrt{x}}=dt
$$
so
$$
\sqrt{x}=\frac{1}{2}(t+c)
$$
and finally
$$
x=\frac{1}{4}(t+c)^2=\frac{1}{4}(t^2+2ct+c^2)
$$
Can you spot your error?

 $\displaystyle x=\frac{1}{4}(t^2+{\Large\color{red}{2}}ct+c^2)$

