P n 1 r Δ t P n 1 1 r Δ t n P The solution of this discrete model is P n 1 r Δ

# P n 1 r δ t p n 1 1 r δ t n p the solution of this

• 12

This preview shows page 4 - 7 out of 12 pages.

. . P n = (1 + r Δ t ) P n - 1 = (1 + r Δ t ) n P 0 The solution of this discrete model is P n = (1 + r Δ t ) n P 0 , which is an exponential growth Joseph M. Mahaffy, h [email protected] i Lecture Notes – Introduction to Differential Equations — (15/47) The Class — Overview The Class... Introduction Applications of Differential Equations Malthusian Growth Example Definitions - What is a Differential Equation? Classification Malthusian Growth 3 Discrete Malthusian Growth : P n +1 = (1 + 0 . t ) P n P 0 = 4 0 2∆ 4∆ 6∆ 8∆ 10∆ 0 2 4 6 8 10 12 n P n Discrete Malthusian Growth Joseph M. Mahaffy, h [email protected] i Lecture Notes – Introduction to Differential Equa — (16/47)

Subscribe to view the full document.

The Class — Overview The Class... Introduction Applications of Differential Equations Malthusian Growth Example Definitions - What is a Differential Equation? Classification Malthusian Growth 4 Malthusian Growth : Let P ( t ) be the population at time t = t 0 + n t and rearrange the model above P n +1 - P n = r tP n P ( t + ∆ t ) - P ( t ) = t · rP ( t ) P ( t + ∆ t ) - P ( t ) t = rP ( t ) Let ∆ t become very small lim t 0 P ( t + ∆ t ) - P ( t ) t = dP ( t ) dt = rP ( t ) , which is a Differential Equation Joseph M. Mahaffy, h [email protected] i Lecture Notes – Introduction to Differential Equations — (17/47) The Class — Overview The Class... Introduction Applications of Differential Equations Malthusian Growth Example Definitions - What is a Differential Equation? Classification Malthusian Growth 5 Solution of Malthusian Growth Model : The Malthusian growth model dP ( t ) dt = rP ( t ) The rate of change of a population is proportional to the population Let c be an arbitrary constant, so try a solution of the form P ( t ) = ce kt Differentiating dP ( t ) dt = cke kt , which if k = r is rP ( t ), so satisfies the differential equation Joseph M. Mahaffy, h [email protected] i Lecture Notes – Introduction to Differential Equa — (18/47) The Class — Overview The Class... Introduction Applications of Differential Equations Malthusian Growth Example Definitions - What is a Differential Equation? Classification Malthusian Growth 6 Solution of Malthusian Growth Model The Malthusian growth model satisfies P ( t ) = ce rt With the initial condition, P ( t 0 ) = P 0 , then the unique solution is P ( t ) = P 0 e r ( t - t 0 ) Malthusian growth is often called exponential growth Joseph M. Mahaffy, h [email protected] i Lecture Notes – Introduction to Differential Equations — (19/47) The Class — Overview The Class... Introduction Applications of Differential Equations Malthusian Growth Example Definitions - What is a Differential Equation? Classification Example: Malthusian Growth 1 Example: Malthusian Growth Consider the Malthusian growth model dP ( t ) dt = 0 . 02 P ( t ) with P (0) = 100 Skip Example Find the solution Determine how long it takes for this population to double Joseph M. Mahaffy, h [email protected] i Lecture Notes – Introduction to Differential Equa — (20/47)
The Class — Overview The Class...

Subscribe to view the full document.

• Fall '08
• staff

### 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