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
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) 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
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 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
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 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
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
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)
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
logo1
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
logo1
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
logo1
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
logo1
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
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)
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)6
y
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
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