Solve differential equation

1. The problem statement, all variables and given/known data

[itex]\frac{dr}{dt}=r(u-r)[/itex]

2. Relevant equations

i rewrote this as [itex]\frac{dr}{dt}-ur=-r^2[/itex]

i think this is like bernoulli's equation



3. 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?
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution

 

please explain how i would solve by separation.
thanks

 

ok so i get
[itex]\frac{ln(r)-ln(r-u)}{u}=t[/itex]

which i then manipulate (or try to at least ) to get:
[itex]
ln\frac{r}{r-u}=ut
[/itex]

if this is correct then i think i can do:
[itex]
\frac{r}{r-u}=e^{ut}
[/itex]

i know something is wrong cos i dont have [itex] r_0 [/itex]

im working through an example in jordan and smith (nonlinear ordinary differential equations)
so i know what the answer should be, im trying to get to it tho.
:(
please please correc this.