differential equations (natural growth and decay)

Question

anonymous · Answer

attachment

ybarrap · Answer

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