One model for the spread of rumors is one where it is assumed that the rate of spread is proportional to the product of the fraction y of the population who have heard the rumor and the fraction who have not heard the rumor. Here y= y(t) is a function of time satisfying
0≤ y(t) ≤ 1 for all times t.
1. Write a differential equation that
is satisfied by y(t). Hint:what does it mean for a quantity to be proportional to another quantity?
*Would it be something like:
dy/dt = k(a/y)(b/y)
k being a proportion constant and a and b are real numbers???*
2.Solve the differential equation from the last part to get a general solution for y(t)
since y is a fraction, I'd go with
dy/dt = ky(1-y)
since the two fractions have to add up to 1
Use partial fractions to do the integral.
To derive the differential equation satisfied by y(t), we start with the given assumption that the rate of spread of rumors is proportional to the product of the fraction of the population who have heard the rumor (y) and the fraction who have not heard the rumor (1 - y).
The term "proportional" means that there exists a constant of proportionality, let's call it k, such that the rate of spread is given by k times the product of y and (1 - y).
Therefore, the differential equation can be written as:
dy/dt = k * y * (1 - y)
This equation says that the rate at which the fraction of the population who have heard the rumor (y) is changing with respect to time (t) is equal to the constant of proportionality (k) times the product of y and (1 - y).
Now, let's solve this differential equation to find a general solution for y(t).
This is a separable differential equation, meaning we can separate the variables and integrate each side separately. Rearranging the equation:
dy / (y*(1 - y)) = k * dt
Taking the integral of both sides:
∫[1/(y*(1-y))] dy = ∫k dt
To integrate the left side, we can use partial fraction decomposition:
1/(y*(1 - y)) = A/y + B/(1 - y)
Multiplying through by y*(1 - y):
1 = A*(1 - y) + B*y
Expanding and collecting terms:
1 = (A - A*y) + (B*y)
Matching coefficients of y:
0 = -Ay + By
Therefore, A = B.
Substitute A = B = 1:
1/(y*(1 - y)) = 1/y + 1/(1 - y)
Now we can integrate:
∫(1/y + 1/(1 - y)) dy = ∫k dt
∫(1/y) dy + ∫(1/(1 - y)) dy = k*t + C
ln|y| - ln|1 - y| = k*t + C
Using properties of logarithms:
ln|y / (1 - y)| = k*t + C
Taking the exponential of both sides:
y / (1 - y) = e^(k*t + C)
At this point, the constant of integration can be absorbed into a new constant of proportionality:
y / (1 - y) = Ce^(k*t)
Next, we rewrite the equation in terms of y:
y = Ce^(k*t) * (1 - y)
Expanding:
y = Ce^(k*t) - Ce^(k*t) * y
Rearranging terms:
y + Ce^(k*t) * y = Ce^(k*t)
Factoring out y:
y * (1 + Ce^(k*t)) = Ce^(k*t)
Dividing both sides by (1 + Ce^(k*t)):
y = Ce^(k*t) / (1 + Ce^(k*t))
This is the general solution for y(t), obtained by solving the differential equation for the spread of rumors.