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Homework Statement
[itex]\frac{dr}{dt}=r(u-r)[/itex]
Homework Equations
i rewrote this as [itex]\frac{dr}{dt}-ur=-r^2[/itex]
i think this is like bernoulli's equation
The Attempt at a Solution
so i let [itex]w=\frac{1}{r}[/itex] so that [itex]r=\frac{1}{w}[/itex]
this gives me
[itex]\frac{dr}{dt}=-1w^{-2}\frac{dw}{dt}[/itex]
so now i have
[itex]-w^{-2}\frac{dw}{dt}-uw^{-1}=-w^{-2}[/itex]
if i divide by [itex]-w^{-2}[/itex]
then i get a linear differential equation
[itex]\frac{dw}{dt}+uw=1[/itex]
the integrating factor is [itex]e^{ut}[/itex]
so i get [itex]w=\frac{1}{u}+w_0e^{-ut}[/itex]
from my definition of [itex]w[/itex] i have [itex]w_0=\frac{1}{r_0}[/itex] and [itex]w=\frac{1}{r}[/itex]
so i get [itex]r=\frac{ur_0}{r_0+ue^{-ut}}[/itex]
is this right?