Autonomous Differential Equations3. Autonomous equations are separable, but ugly integrals and...
Transcript of Autonomous Differential Equations3. Autonomous equations are separable, but ugly integrals and...
![Page 1: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/1.jpg)
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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
![Page 2: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/2.jpg)
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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
![Page 3: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/3.jpg)
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
![Page 4: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/4.jpg)
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
![Page 5: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/5.jpg)
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
![Page 6: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/6.jpg)
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
![Page 7: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/7.jpg)
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
![Page 8: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/8.jpg)
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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
![Page 9: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/9.jpg)
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
![Page 10: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/10.jpg)
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
![Page 11: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/11.jpg)
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
![Page 12: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/12.jpg)
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
![Page 13: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/13.jpg)
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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
![Page 14: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/14.jpg)
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
![Page 15: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/15.jpg)
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
![Page 16: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/16.jpg)
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
![Page 17: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/17.jpg)
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
![Page 18: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/18.jpg)
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
![Page 19: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/19.jpg)
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
![Page 20: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/20.jpg)
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
![Page 21: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/21.jpg)
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
![Page 22: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/22.jpg)
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
![Page 23: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/23.jpg)
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
![Page 24: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/24.jpg)
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
![Page 25: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/25.jpg)
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
![Page 26: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/26.jpg)
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
![Page 27: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/27.jpg)
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
![Page 28: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/28.jpg)
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
![Page 29: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/29.jpg)
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
![Page 30: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/30.jpg)
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
![Page 31: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/31.jpg)
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
![Page 32: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/32.jpg)
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
![Page 33: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/33.jpg)
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
![Page 34: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/34.jpg)
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
![Page 35: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/35.jpg)
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
![Page 36: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/36.jpg)
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
![Page 37: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/37.jpg)
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
![Page 38: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/38.jpg)
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
![Page 39: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/39.jpg)
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
![Page 40: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/40.jpg)
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
![Page 41: 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](https://reader036.fdocuments.us/reader036/viewer/2022071420/6119415d5040210d3358fd92/html5/thumbnails/41.jpg)
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