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Autonomous Differential Equations Bernd Schr oder logo1 Bernd - - PowerPoint PPT Presentation

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Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2) Autonomous Differential Equations Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science Autonomous Differential

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logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Autonomous Differential Equations

Bernd Schr¨

  • der

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Autonomous Differential Equations

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Autonomous Differential Equations

  • 1. A differential equation of the form y′ = F(y) is

autonomous.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Autonomous Differential Equations

  • 1. A differential equation of the form y′ = F(y) is

autonomous.

  • 2. That is, if the right side does not depend on x, the equation

is autonomous.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 5

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Autonomous Differential Equations

  • 1. A differential equation of the form y′ = 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.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 6

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Autonomous Differential Equations

  • 1. A differential equation of the form y′ = 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. The slopes in the direction field will only depend on y.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 7

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Autonomous Differential Equations

  • 1. A differential equation of the form y′ = 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. The slopes in the direction field will only depend on y.
  • 5. Solutions are invariant under horizontal translations.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Equilibrium Solutions of Autonomous Differential Equations

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Equilibrium Solutions of Autonomous Differential Equations

  • 1. Values y0 with F(y0) = 0 give rise to constant solutions

y(x) = y0.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Equilibrium Solutions of Autonomous Differential Equations

  • 1. Values y0 with F(y0) = 0 give rise to constant solutions

y(x) = y0. These solutions are called equilibrium solutions.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Equilibrium Solutions of Autonomous Differential Equations

  • 1. Values y0 with F(y0) = 0 give rise to constant solutions

y(x) = y0. These solutions are called equilibrium solutions.

  • 2. Equilibrium solutions y(x) = y0 are called stable if and
  • nly if solutions near them converge to y(x) = y0.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Equilibrium Solutions of Autonomous Differential Equations

  • 1. Values y0 with F(y0) = 0 give rise to constant solutions

y(x) = y0. These solutions are called equilibrium solutions.

  • 2. Equilibrium solutions y(x) = y0 are called stable if and
  • nly if solutions near them converge to y(x) = y0.

Otherwise they are called unstable.

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

✻

y

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 15

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

✻

y

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

2

✻

y

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

2 4

✻

y

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

y′ > 0, increasing for y < 0

2 4

✻

y

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

y′ > 0, increasing for y < 0

2 4

✻ ✻

y

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

y′ > 0, increasing for y < 0 for 0 < y < 2 y′ < 0, decreasing

2 4

✻ ✻

y

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

❄

y′ > 0, increasing for y < 0 for 0 < y < 2 y′ < 0, decreasing

2 4

✻ ✻

y

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 22

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

❄

y′ < 0, decreasing y′ > 0, increasing for 2 < y < 4 for y < 0 for 0 < y < 2 y′ < 0, decreasing

2 4

✻ ✻

y

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 23

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

❄ ❄

y′ < 0, decreasing y′ > 0, increasing for 2 < y < 4 for y < 0 for 0 < y < 2 y′ < 0, decreasing

2 4

✻ ✻

y

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 24

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

❄ ❄

y′ < 0, decreasing y′ > 0, increasing y′ > 0, increasing for 2 < y < 4 for y < 0 for y > 4 for 0 < y < 2 y′ < 0, decreasing

2 4

✻ ✻

y

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 25

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

❄ ❄

y′ < 0, decreasing y′ > 0, increasing y′ > 0, increasing for 2 < y < 4 for y < 0 for y > 4 for 0 < y < 2 y′ < 0, decreasing

2 4

✻ ✻ ✻

y

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 26

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

❄ ❄

y′ < 0, decreasing y′ > 0, increasing y′ > 0, increasing for 2 < y < 4 for y < 0 for y > 4 for 0 < y < 2 y′ < 0, decreasing

2 4

✻ ✻ ✻

y stable

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 27

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

❄ ❄

y′ < 0, decreasing y′ > 0, increasing y′ > 0, increasing for 2 < y < 4 for y < 0 for y > 4 for 0 < y < 2 y′ < 0, decreasing

2 4

✻ ✻ ✻

y stable unstable

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 28

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

❄ ❄

y′ < 0, decreasing y′ > 0, increasing y′ > 0, increasing for 2 < y < 4 for y < 0 for y > 4 for 0 < y < 2 y′ < 0, decreasing

2 4

✻ ✻ ✻

y stable unstable (semi-stable)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 29

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

❄ ❄

y′ < 0, decreasing y′ > 0, increasing y′ > 0, increasing for 2 < y < 4 for y < 0 for y > 4 for 0 < y < 2 y′ < 0, decreasing

2 4

✻ ✻ ✻

y stable unstable unstable (semi-stable)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 31

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 32

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 33

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 34

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 35

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 36

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 37

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 38

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 39

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 40

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations

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SLIDE 41

logo1 Definition Equilibrium Solutions An Example (Take 1) An Example (Take 2)

Find and Classify the Equilibrium Solutions of y′ = 1 2y(y−2)2(y−4)

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science Autonomous Differential Equations