Autonomous Differential Equations3. Autonomous equations are separable, but ugly integrals and expressions that cannot be solved for y make qualitative analysis sensible. 4. The slopes

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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


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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.



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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.





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

is autonomous.





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is autonomous.3. Autonomous equations are separable, but ugly integrals

and expressions that cannot be solved for y makequalitative analysis sensible.




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

5. Solutions are invariant under horizontal translations.



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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.



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1. Values y0 with F(y0) = 0 give rise to constant solutionsy(x) = y0.

These solutions are called equilibriumsolutions.




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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.



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Find and Classify the Equilibrium Solutions of y′ =12

y(y−2)2(y−4)

6y



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y(y−2)2(y−4)6

y



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y(y−2)2(y−4)

0

6y



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y(y−2)2(y−4)

0

2

6y



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y(y−2)2(y−4)

0

2

4

6y



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y(y−2)2(y−4)

y′ > 0, increasingfor y < 0

0

2

4

6y



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y(y−2)2(y−4)


0

2

4

6

6y



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y(y−2)2(y−4)


for 0 < y < 2y′ < 0, decreasing

0

2

4

6

6y



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y(y−2)2(y−4)

?



0

2

4

6

6y



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y(y−2)2(y−4)

?

y′ < 0, decreasing

y′ > 0, increasing

for 2 < y < 4

for y < 0


0

2

4

6

6y



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y(y−2)2(y−4)

?

?



for 2 < y < 4

for y < 0


0

2

4

6

6y



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y(y−2)2(y−4)

?

?




for 2 < y < 4

for y < 0

for y > 4


0

2

4

6

6y



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y(y−2)2(y−4)

?

?




for 2 < y < 4

for y < 0

for y > 4


0

2

4

6

6

6y



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y(y−2)2(y−4)

?

?




for 2 < y < 4

for y < 0

for y > 4


0

2

4

6

6

6y

stable



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y(y−2)2(y−4)

?

?




for 2 < y < 4

for y < 0

for y > 4


0

2

4

6

6

6y

stable

unstable



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y(y−2)2(y−4)

?

?




for 2 < y < 4

for y < 0

for y > 4


0

2

4

6

6

6y

stable

unstable(semi-stable)



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y(y−2)2(y−4)

?

?




for 2 < y < 4

for y < 0

for y > 4


0

2

4

6

6

6y

stable

unstable

unstable

(semi-stable)



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y(y−2)2(y−4)



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y(y−2)2(y−4)



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y(y−2)2(y−4)



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y(y−2)2(y−4)



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y(y−2)2(y−4)



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y(y−2)2(y−4)



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y(y−2)2(y−4)



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y(y−2)2(y−4)



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y(y−2)2(y−4)



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y(y−2)2(y−4)



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y(y−2)2(y−4)



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y(y−2)2(y−4)



Documents

Autonomous Differential Equations3. Autonomous equations are separable, but ugly integrals and expressions that cannot be solved for y make qualitative analysis sensible. 4. The slopes