I have the following IVP:
$\frac{da}{dt}=ka^{\frac{1}{2}}$
Subject to $a(0)=a_0$, $t\geq0$, $a\geq 0$. This is a separable equation, which I naively returns the solution:
$a(t)=\left(\frac{1}{2}kt + \sqrt{a_0}\right)^2$
However, this is clearly incorrect. For instance, if $a_0=0$, the solution is obviously $a(t)=0$. If $k<0$, the solution has a minimum at $\left(-\frac{2\sqrt{a_0}}{k}, 0\right)$. However, when $a(t)=0$, $a'(t)=0$, so the solution will remain at 0. I'm guessing that the solution is therefore piecewise for $k<0$, but I'm still confused about what's going on with $a_0=0$. The solution surely can't be discontinuous with respect to the initial condition, can it?