Problem - how to find λ?

Question

Guys I need some help with this problem, I tried a thousand times but I couldn't find a realistic answer.

Solve numerically:

x'=-λ.x.y

Where: n=x+y=500; x0= 499 (initial condition); with x=people who ‘do not know’ and y=people who ‘know’; n=total number of people;

Find λ.

λ is the ‘rate of propagation of the rumor’; λ is a constant; The time scale is not important.

Thanks!!

Answer 1

It will take an infinite time for $x(t)$ to go down to zero, so you can introduce a time scale by assuming a "half-life" $\tau$ satisfying $x(\tau)=250$

Then the problem can be solved analytically ...

$$-\lambda \int _0^\tau dt = \int_{499}^{250}\frac{dx}{x(500-x)} $$ You end up with $$ \lambda \tau = \frac{ln(499)}{500} $$

Answer 2

Eliminating $y$ leads to the ODE $$ x'=-λ·x·(500-x). $$ Parametrizing the function as $x(t)=500·u(500λ·t)$ gives the (time-reversed) logistic equation $$ u'(t)=-u(t)·(1-u(t)),\;u(0)=\frac{499}{500} $$ which can now be solved numerically.

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As a Bernoulli equation, one would rather parametrize $x(t)=500/u(500λ·t)$ leading to the linear equation $$ u'(t)=u(t)-1,\; u(0)=\frac{500}{499}. $$