here is my problem and what I have done so far.
$\displaystyle \frac{dx}{dt} = x(1 - x)$
$\displaystyle \frac{dt}{dx} = \frac{1}{x(1 - x)}$.
Now using the method of partial fractions:
$\displaystyle \frac{A}{x} + \frac{B}{1 - x} = \frac{1}{x(1 - x)}$
$\displaystyle \frac{A(1 - x) + Bx}{x(1 - x)} = \frac{1}{x(1 - x)}$
$\displaystyle A(1 - x) + Bx = 1$
$\displaystyle A - Ax + Bx = 1$
$\displaystyle A + (B - A)x = 1 + 0x$.
Therefore $\displaystyle A = 1$ and $\displaystyle B - A = 0$, so $\displaystyle B = 1$.
Thus $\displaystyle \frac{1}{x(1 - x)} = \frac{1}{x} + \frac{1}{1 - x}$.
Back to the DE:
$\displaystyle \frac{dt}{dx} = \frac{1}{x(1 - x)}$
$\displaystyle \frac{dt}{dx} = \frac{1}{x}+ \frac{1}{1 - x}$
$\displaystyle t = \int{\left(\frac{1}{x} + \frac{1}{1 - x}\right)\,dx}$
$\displaystyle t = \ln{|x|} - \ln{|1 - x|} + C$
$\displaystyle t = \ln{\left|\frac{x}{1 - x}\right|} + C$
$\displaystyle t = \ln{\left|\frac{x - 1 + 1}{1 - x}\right|} + C$
$\displaystyle t = \ln{\left|\frac{-(1 - x)}{1 - x} + \frac{1}{1 - x}\right|} + C$
$\displaystyle t = \ln{\left|-1 + \frac{1}{1 - x}\right|} + C$
$\displaystyle t - C = \ln{\left|-1 + \frac{1}{1 - x}\right|}$
$\displaystyle e^{t - C} = \left|-1 + \frac{1}{1 - x}\right|$
$\displaystyle e^{-C}e^t = \left|-1 + \frac{1}{1 - x}\right|$
$\displaystyle \pm e^{-C}e^t = -1 + \frac{1}{1 - x}$
$\displaystyle A\,e^t = -1 + \frac{1}{1 - x}$, where $\displaystyle A = \pm e^{-C}$
$\displaystyle A\,e^t + 1 = \frac{1}{1 - x}$
$\displaystyle \frac{1}{A\,e^t + 1} = 1 - x$
$\displaystyle x = 1 - \frac{1}{A\,e^t + 1}$
$\displaystyle x = \frac{A\,e^t + 1 - 1}{A\,e^t + 1}$
$\displaystyle x = \frac{A\,e^t}{A\,e^t + 1}$.
If we use this DE solution, x=1/(1-e^(-t)) and try to find the proportion of the population that has heard the rumor at the time t=0. The solution gives us the result - 0.5; that is a half of population before the rumor has started spreading? It doesn't make sense. Am I interpreting it wrong?
1. The given solution is x=1/(1+e^(-t)), not what you have said.
2. t = 0 => x = 1/2. All that means is that at t = 0 half the population have heard the rumour. Big deal.
3. The question asked you to show that the solution was x=1/(1+e^(-t)). So you don't actually have to solve the DE. Just substitute x=1/(1+e^(-t)) into it and show that the resulting left hand and right hand sides are equal to each other. The fact that you have been given no boundary condition suggests that this is the approach you were meant to take ... (And given what I have said in my second point, a possible boundary condition would have been the initial condition x(0) = 1/2).