Mahaffy h jmahaffysdsuedu i Lecture Notes Direction Fields and Phase Por 1850

Mahaffy h jmahaffysdsuedu i lecture notes direction

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Joseph M. Mahaffy, h [email protected] i Lecture Notes – Direction Fields and Phase Por — (18/50)
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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 Por — (19/50)
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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 Por — (20/50)
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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 Por — (21/50)
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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 Por — (22/50)
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Mathematical Modeling Introduction to MatLab
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