# ch1 - CHAPTER 1 Section 1 Differential Equation Models 1.1...

• Notes
• 12

This preview shows pages 1–2. Sign up to view the full content.

CHAPTER 1 Section 1. Differential Equation Models 1.1. Let y(t) be the number of bacteria at time t . The rate of change of the number of bacteria is y (t) . Since this rate of change is given to be proportional to y(t) , the resulting differential equation is y (t) = ky(t) . Note that k is a positive constant since y (t) must be positive (i.e. the number of bacteria is growing). 1.2. Let y(t) be the number of field mice at time t . The rate of change of the number of mice is y (t) . Since this rate of change is given to be inversely proportional to the square root of y(t) , the resulting differential equation is y (t) = k/ y(t) . Note that k is a positive constant since y (t) must be positive (i.e. the number of mice is growing). 1.3. Let y(t) be the number of ferrets at time t . The rate of change of the number of ferrets is y (t) . Since this rate of change is given to be proportional to the product of y(t) and the difference between the maximum population and y(t) (i.e. 100 y(t) ), the resulting differential equation is y (t) = ky(t)( 100 y(t)) . Note that k is a positive constant since y (t) must be positive (i.e. the number of ferrets is growing provided y(t) < 100). 1.4. Let y(t) be the quantity of radioactive substance at time t . The rate of change of the material is y (t) . Since this rate of change (decay) is given to be proportional to y(t) , the resulting differential equation is y (t) = − ky(t) . Note that k is a positive constant since y (t) must be negative (i.e. the quantity of radioactive material is decreasing). 1.5. Let y(t) be the quantity of material at time t . The rate of change of the material is y (t) . Since this rate of change (decay) is given to be inversely proportional to y(t) , the resulting differential equation is y (t) = − k/y(t) . Note that k is a positive constant since y (t) must be negative (i.e. the quantity of material is decreasing). 1.6. Let y(t) be the temperature of the potato at time t . The rate of change of the temperature is y (t) . Since this rate of change is given to be proportional to the difference between the potato’s temperature and that of the surrounding room (i.e. y(t) 65), the resulting differential equation is y (t) = − k(y(t) 65 ) . Note that k is a positive constant since y (t) must be negative (i.e. the potato is cooling) and since y(t) 65 > 0 (i.e. the potato is hotter than the surrounding room). 1.7. Let y(t) be the temperature of the thermometer at time t . The rate of change of the temperature is y (t) . Since this rate of change is given to be proportional to the difference between the thermometer’s temperature and that of the surrounding room (i.e. 77 y(t) ), the resulting differential equation is y (t) = k( 77 y(t)) . Note that k is a positive constant since y (t) must be positive (i.e. the thermometer is warming) and since 77 y(t) > 0 (i.e. the thermometer is cooler than the surrounding room).

This preview has intentionally blurred sections. Sign up to view the full version.

This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern