Help with 2 word problems

[mike21]

Hi im wondering if someone could help me solve these two word problems, I really have no idea how to.

1. A scientist reports that a certain strain of bacteria grows at a rate proportional to the square of the size of the population. Set up a differential equation that describes the growth of the population. sketch the solution.

2. At one point in his study of a falling body starting at rest, Galileo conjectured that its velocity at any time is proportional to the distance it has dropped. Using this hypothesis, set up the differential equation whose solution y=f(t), the distance fallen by time t. By making use of the initial value, show why galileos original conjunction is invalid.


Someone please help......

 

[galactus]

2. At one point in his study of a falling body starting at rest, Galileo conjectured that its velocity at any time is proportional to the distance it has dropped. Using this hypothesis, set up the differential equation whose solution y=f(t), the distance fallen by time t. By making use of the initial value, show why galileos original conjunction is invalid.

We know that velocity is the derivative of position, y: \(\displaystyle v=\frac{ds}{dt}\)

Therefore, if velocity is proportional to distance, we have \(\displaystyle \frac{dy}{dt}=ks\)

where k is some constant of proportionality.

Solving this DE, we get \(\displaystyle y=Ce^{kt}\)

Can you find the flaw in the logic?.