(adsbygoogle = window.adsbygoogle || []).push({}); 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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Solve differential equation

**Physics Forums | Science Articles, Homework Help, Discussion**