Science:Math Exam Resources/Courses/MATH100 B/December 2023/Question 20/Solution 1

We know that if the function ${\textstyle y}$ is at an equilibrium point, then it is not changing, which means the derivative of the function is zero. Since we are given the differential equation in factored form, we can read off the zeros easily: ${\textstyle y'=0}$ when ${\textstyle y=0,2,10}$; these are the equilibrium points. To determine if the function is increasing or decreasing on either side of the equilibrium points, we look at whether ${\textstyle y'}$ is positive or negative.

When ${\textstyle y<0}$, ${\textstyle y'}$ is positive. When ${\textstyle 0<y<2}$, ${\textstyle y'}$ is negative. When ${\textstyle 2<y<10}$, ${\textstyle y'}$ is positive. And when ${\textstyle y>10}$, ${\textstyle y'}$ is negative. So, we have that the function is increasing for ${\textstyle y<0}$, decreasing for ${\textstyle 0<y<2}$, increasing for ${\textstyle 2<y<10}$, and decreasing for ${\textstyle y>10}$, which means ${\textstyle y=0}$ is an unstable equilibrium, ${\textstyle y=2}$ is a stable equilibrium, and ${\textstyle y=10}$ is an unstable equilibrium. The phase line representing all of this is given below.

math100 B 2023 Q20 solution: phase diagram