Equilibria & Stability — Practice Problems and Worked Solutions

17 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Classify the equilibrium.

   $$y^{\prime }=y+1,y_{\ast }=-1$$

   Answer: $\text{Unstable}$

   Step-by-step solution

   1. $$f^{\prime }(y_{\ast })=1$$
   2. $$\text{Unstable}$$
2. 2.

   Classify the equilibrium.

   $$y^{\prime }=2y-2,y_{\ast }=1$$

   Answer: $\text{Unstable}$

   Step-by-step solution

   1. $$f^{\prime }(y_{\ast })=2$$
   2. $$\text{Unstable}$$
3. 3.

   Classify the equilibrium.

   $$y^{\prime }=2-y,y_{\ast }=2$$

   Answer: $\text{Asymptotically stable}$

   Step-by-step solution

   1. $$f^{\prime }(y_{\ast })=-1$$
   2. $$\text{Asymptotically stable}$$
4. 4.

   Classify the equilibrium.

   $$y^{\prime }=-2y-2,y_{\ast }=-1$$

   Answer: $\text{Asymptotically stable}$

   Step-by-step solution

   1. $$f^{\prime }(y_{\ast })=-2$$
   2. $$\text{Asymptotically stable}$$
5. 5.

   Classify the equilibrium.

   $$y^{\prime }=3y,y_{\ast }=0$$

   Answer: $\text{Unstable}$

   Step-by-step solution

   1. $$f^{\prime }(y_{\ast })=3$$
   2. $$\text{Unstable}$$
6. 6.

   Classify the equilibrium.

   $$y^{\prime }=3-3y,y_{\ast }=1$$

   Answer: $\text{Asymptotically stable}$

   Step-by-step solution

   1. $$f^{\prime }(y_{\ast })=-3$$
   2. $$\text{Asymptotically stable}$$
7. 7.

   Classify the equilibrium.

   $$y^{\prime }=4y-8,y_{\ast }=2$$

   Answer: $\text{Unstable}$

   Step-by-step solution

   1. $$f^{\prime }(y_{\ast })=4$$
   2. $$\text{Unstable}$$
8. 8.

   Classify the equilibrium.

   $$y^{\prime }=12-4y,y_{\ast }=3$$

   Answer: $\text{Asymptotically stable}$

   Step-by-step solution

   1. $$f^{\prime }(y_{\ast })=-4$$
   2. $$\text{Asymptotically stable}$$
9. 9.

   Classify the equilibrium.

   $$y^{\prime }=5y+10,y_{\ast }=-2$$

   Answer: $\text{Unstable}$

   Step-by-step solution

   1. $$f^{\prime }(y_{\ast })=5$$
   2. $$\text{Unstable}$$
10. 10.

    Classify the equilibrium.

    $$y^{\prime }=-5y,y_{\ast }=0$$

    Answer: $\text{Asymptotically stable}$

    Step-by-step solution

    1. $$f^{\prime }(y_{\ast })=-5$$
    2. $$\text{Asymptotically stable}$$
11. 11.

    Classify the equilibrium.

    $$y^{\prime }=y-3,y_{\ast }=3$$

    Answer: $\text{Unstable}$

    Step-by-step solution

    1. $$f^{\prime }(y_{\ast })=1$$
    2. $$\text{Unstable}$$
12. 12.

    Classify the equilibrium.

    $$y^{\prime }=-y-3,y_{\ast }=-3$$

    Answer: $\text{Asymptotically stable}$

    Step-by-step solution

    1. $$f^{\prime }(y_{\ast })=-1$$
    2. $$\text{Asymptotically stable}$$
13. 13.

    Classify the equilibrium.

    $$y^{\prime }=2y+8,y_{\ast }=-4$$

    Answer: $\text{Unstable}$

    Step-by-step solution

    1. $$f^{\prime }(y_{\ast })=2$$
    2. $$\text{Unstable}$$
14. 14.

    Classify the equilibrium.

    $$y^{\prime }=10-2y,y_{\ast }=5$$

    Answer: $\text{Asymptotically stable}$

    Step-by-step solution

    1. $$f^{\prime }(y_{\ast })=-2$$
    2. $$\text{Asymptotically stable}$$
15. 15.

    Classify the equilibrium.

    $$y^{\prime }=6y-6,y_{\ast }=1$$

    Answer: $\text{Unstable}$

    Step-by-step solution

    1. $$f^{\prime }(y_{\ast })=6$$
    2. $$\text{Unstable}$$
16. 16.

    Classify the equilibrium.

    $$y^{\prime }=12-6y,y_{\ast }=2$$

    Answer: $\text{Asymptotically stable}$

    Step-by-step solution

    1. $$f^{\prime }(y_{\ast })=-6$$
    2. $$\text{Asymptotically stable}$$
17. 17.

    Classify the equilibrium.

    $$y^{\prime }=3y-12,y_{\ast }=4$$

    Answer: $\text{Unstable}$

    Step-by-step solution

    1. $$f^{\prime }(y_{\ast })=3$$
    2. $$\text{Unstable}$$

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