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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?