Autonomous Differential Equations

1 Logo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2) Autonomous Differential EquationsBernd Schr oderBernd Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2) Autonomous Differential Differential equation of the formy =F(y) is, if the right side does not depend onx, the equationis Equations are separable, but ugly integralsand expressions that cannot be solved forymakequalitative analysis slopes in the direction field will only depend are invariant under horizontal Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2) Autonomous Differential Differential equation of the formy =F(y)

2 Is, if the right side does not depend onx, the equationis Equations are separable, but ugly integralsand expressions that cannot be solved forymakequalitative analysis slopes in the direction field will only depend are invariant under horizontal Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2) Autonomous Differential Differential equation of the formy =F(y) is, if the right side does not depend onx, the equationis Equations are separable, but ugly integralsand expressions that cannot be solved forymakequalitative analysis slopes in the direction field will only depend are invariant under horizontal Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2) Autonomous Differential Differential equation of the formy =F(y)

3 Is, if the right side does not depend onx, the equationis Equations are separable, but ugly integralsand expressions that cannot be solved forymakequalitative analysis slopes in the direction field will only depend are invariant under horizontal Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2) Autonomous Differential Differential equation of the formy =F(y) is, if the right side does not depend onx, the equationis Equations are separable, but ugly integralsand expressions that cannot be solved forymakequalitative analysis slopes in the direction field will only depend are invariant under horizontal Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2) Autonomous Differential Differential equation of the formy =F(y)

4 Is, if the right side does not depend onx, the equationis Equations are separable, but ugly integralsand expressions that cannot be solved forymakequalitative analysis slopes in the direction field will only depend are invariant under horizontal Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Equilibrium Solutions of AutonomousDifferential (y0)=0 give rise to constant solutionsy(x)= solutions are solutionsy(x)=y0are calledstableif andonly if solutions near them converge toy(x)= they are Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Equilibrium Solutions of AutonomousDifferential (y0)=0 give rise to constant solutionsy(x)= solutions are solutionsy(x)=y0are calledstableif andonly if solutions near them converge toy(x)= they are Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Equilibrium Solutions of AutonomousDifferential (y0)=0 give rise to constant solutionsy(x)= solutions are solutionsy(x)

5 =y0are calledstableif andonly if solutions near them converge toy(x)= they are Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Equilibrium Solutions of AutonomousDifferential (y0)=0 give rise to constant solutionsy(x)= solutions are solutionsy(x)=y0are calledstableif andonly if solutions near them converge toy(x)= they are Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Equilibrium Solutions of AutonomousDifferential (y0)=0 give rise to constant solutionsy(x)= solutions are solutionsy(x)=y0are calledstableif andonly if solutions near them converge toy(x)= they are Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Find and Classify the Equilibrium Solutions ofy =12y(y 2)2(y 4)6yBernd Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Find and Classify the Equilibrium Solutions ofy =12y(y 2)2(y 4)

6 6yBernd Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Find and Classify the Equilibrium Solutions ofy =12y(y 2)2(y 4)06yBernd Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Find and Classify the Equilibrium Solutions ofy =12y(y 2)2(y 4)026yBernd Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Find and Classify the Equilibrium Solutions ofy =12y(y 2)2(y 4)0246yBernd Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Find and Classify the Equilibrium Solutions ofy =12y(y 2)2(y 4)y >0, increasingfory<00246yBernd Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Find and Classify the Equilibrium Solutions ofy =12y(y 2)2(y 4)y >0, increasingfory<002466yBernd Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)

7 Find and Classify the Equilibrium Solutions ofy =12y(y 2)2(y 4)y >0, increasingfory<0for 0<y<2y <0, decreasing02466yBernd Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Find and Classify the Equilibrium Solutions ofy =12y(y 2)2(y 4)?y >0, increasingfory<0for 0<y<2y <0, decreasing02466yBernd Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Find and Classify the Equilibrium Solutions ofy =12y(y 2)2(y 4)?y <0, decreasingy >0, increasingfor 2<y<4fory<0for 0<y<2y <0, decreasing02466yBernd Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Find and Classify the Equilibrium Solutions ofy =12y(y 2)2(y 4)?

8 ?y <0, decreasingy >0, increasingfor 2<y<4fory<0for 0<y<2y <0, decreasing02466yBernd Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Find and Classify the Equilibrium Solutions ofy =12y(y 2)2(y 4)??y <0, decreasingy >0, increasingy >0, increasingfor 2<y<4fory<0fory>4for 0<y<2y <0, decreasing02466yBernd Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Find and Classify the Equilibrium Solutions ofy =12y(y 2)2(y 4)??y <0, decreasingy >0, increasingy >0, increasingfor 2<y<4fory<0fory>4for 0<y<2y <0, decreasing024666yBernd Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Find and Classify the Equilibrium Solutions ofy =12y(y 2)2(y 4)?

9 ?y <0, decreasingy >0, increasingy >0, increasingfor 2<y<4fory<0fory>4for 0<y<2y <0, decreasing024666ystableBernd Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Find and Classify the Equilibrium Solutions ofy =12y(y 2)2(y 4)??y <0, decreasingy >0, increasingy >0, increasingfor 2<y<4fory<0fory>4for 0<y<2y <0, decreasing024666ystableunstableBernd Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Find and Classify the Equilibrium Solutions ofy =12y(y 2)2(y 4)??y <0, decreasingy >0, increasingy >0, increasingfor 2<y<4fory<0fory>4for 0<y<2y <0, decreasing024666ystableunstable(semi-sta ble)Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Find and Classify the Equilibrium Solutions ofy =12y(y 2)2(y 4)?

10 ?y <0, decreasingy >0, increasingy >0, increasingfor 2<y<4fory<0fory>4for 0<y<2y <0, decreasing024666ystableunstableunstable( semi-stable)Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Find and Classify the Equilibrium Solutions ofy =12y(y 2)2(y 4)Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Find and Classify the Equilibrium Solutions ofy =12y(y 2)2(y 4)Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Find and Classify the Equilibrium Solutions ofy =12y(y 2)2(y 4)Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn Example (Take 1)An Example (Take 2)Find and Classify the Equilibrium Solutions ofy =12y(y 2)2(y 4)Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceAutonomous Differential Equationslogo1 DefinitionEquilibrium SolutionsAn E