# Autonomous Differential Equations

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DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Autonomous Differential EquationsBernd SchröderBernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Autonomous Differential EquationsBernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Autonomous Differential Equations1. A differential equation of the form y0 F(y) isautonomous.Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Autonomous Differential Equations1. A differential equation of the form y0 F(y) isautonomous.2. That is, if the right side does not depend on x, the equationis autonomous.Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Autonomous Differential Equations1. A differential equation of the form y0 F(y) isautonomous.2. That is, if the right side does not depend on x, the equationis autonomous.3. Autonomous equations are separable, but ugly integralsand expressions that cannot be solved for y makequalitative analysis sensible.Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Autonomous Differential Equations1. A differential equation of the form y0 F(y) isautonomous.2. That is, if the right side does not depend on x, the equationis autonomous.3. Autonomous equations are separable, but ugly integralsand expressions that cannot be solved for y makequalitative analysis sensible.4. The slopes in the direction field will only depend on y.Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Autonomous Differential Equations1. A differential equation of the form y0 F(y) isautonomous.2. That is, if the right side does not depend on x, the equationis autonomous.3. Autonomous equations are separable, but ugly integralsand expressions that cannot be solved for y makequalitative analysis sensible.4. The slopes in the direction field will only depend on y.5. Solutions are invariant under horizontal translations.Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Equilibrium Solutions of AutonomousDifferential EquationsBernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Equilibrium Solutions of AutonomousDifferential Equations1. Values y0 with F(y0 ) 0 give rise to constant solutionsy(x) y0 .Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Equilibrium Solutions of AutonomousDifferential Equations1. Values y0 with F(y0 ) 0 give rise to constant solutionsy(x) y0 . These solutions are called equilibriumsolutions.Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Equilibrium Solutions of AutonomousDifferential Equations1. Values y0 with F(y0 ) 0 give rise to constant solutionsy(x) y0 . These solutions are called equilibriumsolutions.2. Equilibrium solutions y(x) y0 are called stable if andonly if solutions near them converge to y(x) y0 .Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Equilibrium Solutions of AutonomousDifferential Equations1. Values y0 with F(y0 ) 0 give rise to constant solutionsy(x) y0 . These solutions are called equilibriumsolutions.2. Equilibrium solutions y(x) y0 are called stable if andonly if solutions near them converge to y(x) y0 .Otherwise they are called unstable.Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2y6Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2y60Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2y620Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2y6420Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2y6420Bernd SchröderAutonomous Differential Equationsy0 0, increasingfor y 0logo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2y6420y0 0, increasingfor y 06Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2y6420y0 0, decreasingfor 0 y 2y0 0, increasingfor y 06Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2y6420y0 0, decreasing?for 0 y 2y0 0, increasingfor y 06Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2y6420y0 0, decreasingfor 2 y 4y0 0, decreasing?for 0 y 2y0 0, increasingfor y 06Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2y6420y0 0, decreasing?for 2 y 4y0 0, decreasing?for 0 y 2y0 0, increasingfor y 06Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2y 60y 0, increasingfor y 4420y0 0, decreasing?for 2 y 4y0 0, decreasing?for 0 y 2y0 0, increasingfor y 06Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2y 60y 0, increasing6for y 4420y0 0, decreasing?for 2 y 4y0 0, decreasing?for 0 y 2y0 0, increasingfor y 06Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2y 60y 0, increasing6for y 442stable 0y0 0, decreasing?for 2 y 4y0 0, decreasing?for 0 y 2y0 0, increasingfor y 06Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2y 60y 0, increasing6for y 44unstable 2stable 0y0 0, decreasing?for 2 y 4y0 0, decreasing?for 0 y 2y0 0, increasingfor y 06Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2y 60y 0, increasing6for y 44unstable 2(semi-stable)stable 0y0 0, decreasing?for 2 y 4y0 0, decreasing?for 0 y 2y0 0, increasingfor y 06Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2y 60y 0, increasing6for y 4unstable 4unstable 2(semi-stable)stable 0y0 0, decreasing?for 2 y 4y0 0, decreasing?for 0 y 2y0 0, increasingfor y 06Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)1Find and Classify the Equilibrium Solutions of y0 y(y 2)2 (y 4)2Bernd SchröderAutonomous Differential Equationslogo1Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations 1. A differential equation of the form y0 F(y) is autonomous. 2. That is, if the right side does not depend on x, the equation is autonomous. 3. Autonomous equations are separable, but ugly integrals and expressions that cannot be solved for y make qualitative analysis sensible. 4.

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