The given model for the spread of a rumor can be represented by the following differential equation:
dy/dt = ky(1 - y)
where:
- dy/dt represents the rate of change of the fraction of the population who have heard the rumor over time t.
- k is the constant of proportionality.
- y represents the fraction of the population who have heard the rumor.
To solve this differential equation, we can use separation of variables.
Separate the variables:
dy / (y(1 - y)) = k dt
Integrate both sides:
∫ (1 / (y(1 - y))) dy = ∫ k dt
The integral on the left side can be rewritten using partial fraction decomposition.
1 / (y(1 - y)) = A/y + B/(1 - y)
Multiplying both sides by y(1 - y), we get:
1 = A(1 - y) + By
Let's solve for A and B:
1 = A - Ay + By
1 = A + (B - A)y
This equation must hold for all values of y. Therefore, the coefficients must be equal:
A = 1
B - A = 0 ⟹ B = A = 1
Now, integrate both sides:
∫ (1/y + 1/(1 - y)) dy = ∫ k dt
ln|y| - ln|1 - y| = kt + C
ln|y / (1 - y)| = kt + C
Apply exponentiation:
(y / (1 - y)) = e^(kt + C)
Now, solve for y:
y = (e^(kt + C)) / (1 + e^(kt + C))
Given y(0) = y0:
y0 = (e^C) / (1 + e^C)
Solve for C:
e^C = (y0) / (1 - y0)
Plug back into the solution for y:
y = ((y0 e^(kt)) / (1 - y0 + y0 e^(kt)))
So, the solution to the differential equation is:
y(t) = ((y0 e^(kt)) / (1 - y0 + y0 e^(kt)))
