Chapter 8 - Mathematical Modeling With Differential Equations - 8.1 Modeling With Differential Equations - Exercises Set 8.1 - Page 567: 27

$(a) \frac{dy}{dt} = ky^2, y(0) = y_o, k>0$ $(b) \frac{dy}{dt} = -ky^2, y(0) = y_o, k>0$

$(a)$The sentence "increases at a rate that is proportional to the square of the amount present" is clearly $y^2$ with constant k $\frac{dy}{dt} = ky^2, y(0) = y_o, k>0$ $(b)$ The sentence "decreases at a rate that is proportional to the square of the amount present" means $-y^2$ with constant k $\frac{dy}{dt} = -ky^2, y(0) = y_o, k>0$