Simple seperable first-order equation

[Tracker]

I am trying to solve [math] \frac{dx}{dt} = Rx(t)(1 - \frac{x(t)}{K} [/math] knowing [math] x(0) = K/N [/math]

 

I got to [math] dx = [Rx(t) - R(x(t))^2]dt [/math] and then I am unsure about how I should do the integration of the second term of the right hand side.

 

The solution is [math] x(t) = \frac{K}{1 + (N - 1)e^(-Rt)}[/math]

[Schrödinger's hat]

Well first of all get the functions of x on LHS otherwise you can't integrate (cannot integrate x w/ respect to t unless you know what it is):

[math]\int\frac{1}{Rx\left(1-\frac{x}{K}\right)}\frac{dx}{dt}dt=\int 1 dt[/math]

 

Then it looks like partial fractions will work so find A, B st.

[math]\int\frac{1}{Rx\left(1-\frac{x}{K}\right)}dt=\frac{1}{R}\int \frac{A}{x}+\frac{B}{1-\frac{x}{K}} dt[/math]

Then you can integrate term by term.

If you need help/explanation for how to find A and B give me another bell.

Edited March 8, 2011 by Schrödinger's hat