Hi, I have a quick question to do with Differential equations.
So i have done the first part of my question no problems but for this next part it says:
Show x(t) is always non-decreasing for t=> 0, without solving the equation.
dx/dt = rx(1-x), where r is just a constant (rate of maximum growth in this case).
I get that it must have something to do with the derivative always being positive but i'm just not sure how I can show this when there are no t variables in the equation.
ALSO, I have another question with finding the general solution to this equation. I have gotten it as far as:
ln(x)-ln(1-x)=rt+c
ln(x/(1-x))=rt+c
x/(1-x)=e^(rt+c)
And I am positive it is correct to this point, however I just can't seem to finish it off and get it in the form x(t)= ......
Thanks for any help!!!
Actually you are wrong on two counts.
First
so you have a sign problem.
Also
So you have
or
Now multiply out and get everything on one side:
Now use the quadratic formula. Go get 'em tiger.
One slight simplification here: If we define for any A' not equal to 0 the equation is much simpler looking.
-Dan
Thanks for giving me a hand! So isn't the integral of 1/(1-x) = -ln(1-x), by using substitution? I think i might have arrived at the solution with it being:
x(t) = Ae^rt/ (1+Ae^(rt)) where A=e^c
And with your first answer, how can i show that the derivative is positive though if x is a function of t, but instead of it actually being in terms of t its just written as x(t), meaning that we don't actually know that when t is positive that x(t) is positive? Ie. if x(t)=1000-t, for all t<1000, won't the derivate be negative?
Also how do you write with the integrals and everything? People always answer my questions so neatly and mine is just a mess trying to write without the maths stuff!! haha
Ahem! Okay, I missed the extra sign. Good catch. But you still need the absolute value!
You proved from the original differential equation that the slope is always positive. The derivative of the explicit form will also show you the same.
The solution looks good. As for the LaTeX we have a subforum.