The graph is given by 10 20 30 40 50 10 10 20 30 t y t Joseph M Mahaffy h

The graph is given by 10 20 30 40 50 10 10 20 30 t y

This preview shows page 5 - 7 out of 13 pages.

The graph is given by 0 10 20 30 40 50 -10 0 10 20 30 t y ( t ) Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Port — (18/50) Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Introduction to MatLab How do we make the previous graph? MatLab is a powerful software for mathematics, engineering, and the sciences MatLab stands for Matrix Laboratory Designed for easy managing of vectors, matrices, and graphics Valuable subroutines and packages for specialty applications It is a necessary tool for anyone in Applied Mathematics Introduction to MatLab Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Portraits - 1D — (19/50) Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Example: Logistic Growth Example: Sine Function Autonomous Differential Equation The general first order differential equation satisfies dy dt = f ( t, y ) . A very important set of DEs that we study are called Autonomous Differential Equations Definition (Autonomous Differential Equation) A first order autonomous differential equation has the form dy dt = f ( y ) . The function, f , depends only on the dependent variable. Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Port — (20/50)
Image of page 5

Subscribe to view the full document.

Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Example: Logistic Growth Example: Sine Function Qualitative Behavior of Differential Equations The first step of any qualitative analysis is finding equilibrium solutions Definition (Equilibrium Solutions) Consider autonomous DE dy dt = f ( y ) . If y ( t ) = c is a constant solution or equilibrium solution to this DE, then dy dt = 0. Therefore the constant c is a solution of the algebraic equation f ( y ) = 0 . Equilibrium solutions are also referred to as fixed points , stationary points , or critical points . Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Portraits - 1D — (21/50) Mathematical Modeling Introduction to MatLab Qualitative Behavior of Differential Equations More Examples Maple - Direction Fields Example: Logistic Growth Example: Sine Function Classification of Equilibria There are a variety of local behaviors near an equilibrium, y e 1 An asymptotically stable equilibrium , often referred to as an attractor or sink has any nearby solution approach y e as t → ∞ 2 An unstable equilibrium , often referred to as a repeller or source has any nearby solution leave a region about y e as t → ∞ 3 A neutrally stable equilibrium has any solution stay nearby the equilibrium, but not approach the equilibrium y e as t → ∞ 4 A semi-stable equilibrium (in 1D) has solutions on one side of y e approach y e as t → ∞ , while solutions on the other side of y e diverge away from y e Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Port — (22/50)
Image of page 6
Image of page 7
  • Fall '08
  • staff

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes