Solve first 1st order differential dV/dt=S-CV^1/2

Question

I am trying to solve the following first order differential equation:

$\frac{dV}{dt} = S-CV^\frac{1}{2}$

I can't see how this could be solved via separation of variables or integrating factor as it is not linear.

I don't know if I'm missing something, is there a straight forward way of solving this?

Answer 1

Hint: $$\dfrac{dV}{dt} = S-CV^\frac{1}{2}$$ $$\dfrac{dV}{S-CV^\frac{1}{2}} = dt$$

The integral on the right side is equal to:

$$ -\frac{2 (S \log(S-C \sqrt{V})+C \sqrt{V})}{C^2}+c$$

Answer 2

let $$V^{\frac{1}{2}}=u$$ derive both sides respect to $t$ $$\frac{1}{2}V^{-\frac{1}{2}}\frac{dV}{dt}=\frac{du}{dt}$$ $$\frac{1}{2u}\frac{dV}{dt}=\frac{du}{dt}$$ $$\frac{dV}{dt}=2u\frac{du}{dt}$$ substitute it $$2u\frac{du}{dt}=S-Cu$$ $$\frac{2udu}{S-Cu}=dt$$ $$-\frac{2du}{C}+\frac{2Sdu}{C(S-Cu)}=dt$$