Having bit of trouble with simple simplification Hi there I am having a stump onto how in my book an equation goes from
$$\frac{y}{1-(y/K)}=Ce^{rt}$$ with $$C=\frac{y_{o}}{1-(y_{o}/K)}$$
to solving for y to be
$$y= \frac{y_{o}K}{y_{o}+(K-y_{o})e^{-rt}}$$
I know this is a very trivial question for most, but I am just wondering the best way to do this, because when I tried to derive it I kept getting different terms and such and it was hard to get a nice formula like this.
Thanks
 A: Starting with $$\frac{y}{1-(y/K)}=Ce^{rt}$$ you can flip each side to get $$\frac{1-(y/K)}{y}=\frac{e^{-rt}}{C} \\ \implies \frac{1}{y}-\frac{y}{Ky}=\frac{e^{-rt}}{C}  \\ \implies \frac{1}{y}-\frac{1}{K}=\frac{e^{-rt}}{C}  \\  \implies \frac{1}{y}=\frac{e^{-rt}}{C} +\frac{1}{K} \\ \implies \frac{1}{y}=\frac{Ke^{-rt}+C}{CK} \\ \implies y=\frac{KC}{Ke^{-rt}+C}$$ Now substituting in $C = \frac{y_{o}}{1-(y_{o}/K)}$ yields $$y=\frac{K\frac{y_{o}}{1-(y_{o}/K)}}{Ke^{-rt}+\frac{y_{o}}{1-(y_{o}/K)}} \\ = \frac{Ky_0}{(1-(y_{o}/K))\left(Ke^{-rt}+\frac{y_{o}}{1-(y_{o}/K)}\right)} \\ = \frac{Ky_0}{(1-(y_{o}/K))Ke^{-rt}+y_0\frac{1-(y_{o}/K)}{1-(y_{o}/K)}} \\ = \frac{Ky_0}{(1-(y_{o}/K))Ke^{-rt}+y_0} \\ = \frac{Ky_0}{Ke^{-rt}-Ke^{-rt}(y_{o}/K)+y_0}  \\ = \frac{Ky_0}{Ke^{-rt}-y_0e^{-rt}+y_0}\\ = \frac{Ky_0}{e^{-rt}(K-y_0)+y_0}$$ 
A: $$\frac{y}{1-(y/K)}=Ce^{rt} \iff\frac{Ky}{1-y}=Ce^{rt} \iff $$ 
$$
Ky=Ce^{rt}(K-y) \iff y= \dfrac{KCe^{rt}}{K+Ce^{rt}}=\dfrac{KC}{Ke^{-rt}+C}
$$
For: $$C=\frac{y_{o}}{1-(y_{o}/K)}=\dfrac{Ky_0}{K-y_0}$$
becomes:
$$
y= \dfrac{K^2y_0}{K\left[e^{-rt}(K-y_0)+y_0\right]}=\dfrac{Ky_0}{e^{-rt}(K-y_0)+y_0}
$$
