Autonomous Differential EquationsAutonomous Differential Equations 1. A differential equation of the...

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

Autonomous Differential Equations

Bernd Schroder

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

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

Autonomous Differential Equations

1. A differential equation of the form y′ = 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 Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

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

Autonomous Differential Equations1. A differential equation of the form y′ = F(y) is

autonomous.

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 Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

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

Autonomous Differential Equations1. 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 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 Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

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

Autonomous Differential Equations1. 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 makequalitative analysis sensible.

4. The slopes in the direction field will only depend on y.5. Solutions are invariant under horizontal translations.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Autonomous Differential Equations1. 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 makequalitative analysis sensible.

4. The slopes in the direction field will only depend on y.

5. Solutions are invariant under horizontal translations.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Autonomous Differential Equations1. 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 makequalitative analysis sensible.

4. The slopes in the direction field will only depend on y.5. Solutions are invariant under horizontal translations.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

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

Equilibrium Solutions of AutonomousDifferential Equations

1. 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 Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

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

Equilibrium Solutions of AutonomousDifferential Equations

1. 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 Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

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

Equilibrium Solutions of AutonomousDifferential Equations

1. 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 Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

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

Equilibrium Solutions of AutonomousDifferential Equations

1. 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 Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

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

Equilibrium Solutions of AutonomousDifferential Equations

1. 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 Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

6y

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)6

y

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

0

6y

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

0

2

6y

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

0

2

4

6y

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

y′ > 0, increasingfor y < 0

0

2

4

6y

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

y′ > 0, increasingfor y < 0

0

2

4

6

6y

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

y′ > 0, increasingfor y < 0

for 0 < y < 2y′ < 0, decreasing

0

2

4

6

6y

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

?

y′ > 0, increasingfor y < 0

for 0 < y < 2y′ < 0, decreasing

0

2

4

6

6y

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

?

y′ < 0, decreasing

y′ > 0, increasing

for 2 < y < 4

for y < 0

for 0 < y < 2y′ < 0, decreasing

0

2

4

6

6y

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

?

?

y′ < 0, decreasing

y′ > 0, increasing

for 2 < y < 4

for y < 0

for 0 < y < 2y′ < 0, decreasing

0

2

4

6

6y

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(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 < 2y′ < 0, decreasing

0

2

4

6

6y

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(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 < 2y′ < 0, decreasing

0

2

4

6

6

6y

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(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 < 2y′ < 0, decreasing

0

2

4

6

6

6y

stable

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(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 < 2y′ < 0, decreasing

0

2

4

6

6

6y

stable

unstable

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(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 < 2y′ < 0, decreasing

0

2

4

6

6

6y

stable

unstable(semi-stable)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(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 < 2y′ < 0, decreasing

0

2

4

6

6

6y

stable

unstable

unstable

(semi-stable)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations

logo1

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

Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Autonomous Differential Equations