Question

differential equations (natural growth and decay)

Answer 1

$$
\cfrac{dy(t)}{dt}=ky(t)^{c+1}, y(0)=y_0\\
\int \cfrac{dy(t)}{dt}y(t)^{-c-1}dt=\int k~dt\\
-\cfrac{y(t)}{cy(t)^{c+1}}=kt+c_1\\
-\cfrac{1}{cy(t)^c}=kt+c_1\\
y(t)=\left (\cfrac{-1}{c(kt+c_1)}\right )^{1/c}\\
y(0)=y_0=\left (\cfrac{-1}{c\times c_1)}\right )^{1/c}\\
y_0^c=\cfrac{-1}{c\times c_1}\\
c_1=\cfrac{-1}{c\times y_0^c}\\
y(t)=\left (\cfrac{-1}{c(kt+\cfrac{-1}{c\times y_0^c})}\right )^{1/c}\\
y(t)=\left (\cfrac{-1}{ckt-\cfrac{1}{ y_0^c})}\right )^{1/c}\\
=\left (\cfrac{1}{\cfrac{1}{ y_0^c}-ckt)}\right )^{1/c}\\
$$
That's it!

Make sense?