I'm trying to find an explicit solution $u(t)$ of $$ \dot{u} = \frac{\beta}{\sqrt{\alpha t}-u},\quad u(0)=0 $$ where $\alpha>0$ and $\beta\in\mathbb{R}$ are given constants. I did not know how to solve it directly, so I tried to solve it numerically, but still I could not find a way to find an explicit solution.
Does anybody have idea to solve it. Any explanation would be greatly appreciated.
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With the change of variables $u = v \sqrt{t}$ your ode becomes
$$\dot u = \dot v \sqrt{t} + \frac{v}{2\sqrt{t}} = \frac{\beta}{\sqrt{t}(\sqrt{\alpha} - v)}$$ hence $$ \dot v = \frac{\beta}{t(\sqrt{\alpha} - v)}- \frac{v}{2t} = \frac{1}{t}\left(\frac{\beta}{(\sqrt{\alpha} - v)}- v\right)$$ which is separable.
Change variable to $y=\sqrt{\alpha t}-u$. This will give $$ \frac{dy}{dt}=-\frac{du}{dt}-\sqrt{\alpha}\frac{1}{2\sqrt{t}}. $$ The equation becomes $$ \frac{dy}{dt}=\frac{\beta}{y}-\sqrt{\alpha}\frac{1}{2\sqrt{t}}. $$ This equation can be solved by taking $y=C\sqrt{t}$ that gives $$ \frac{C}{2}-\frac{\beta}{C}=\frac{\sqrt{\alpha}}{2} $$ that determines $C$.