Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

14.16: Section 4.4 Answers

( \newcommand{\kernel}{\mathrm{null}\,}\)

1. ¯y=0 0 is a stable equilibrium; trajectories are v2+y24=c

2. ¯y=0 0 is an unstable equilibrium; trajectories are v2+2y33=c

3. ¯y=0 0 is a stable equilibrium; trajectories are v2+2|y|33=c

4. ¯y=0 0 is a stable equilibrium; trajectories are v2ey(y+1)=c

5. equilibria: 0 (stable) and 2,2 (unstable); trajectories: 2v2y4+8y2=c; separatrix: 2v2y4+8y2=16

6. equilibria: 0 (unstable) and 2,2 (stable); trajectories: 2v2+y48y2=c; separatrix: 2v2+y48y2=0

7. equilibria: 0,2,2 (stable), 1,1 (unstable); trajectories: 6v2+y2(2y415y2+24)=c; separatrix: 6v2+y2(2y415y2+24)=11

8. equilibria: 0,2 (stable) and 2,1 (unstable); trajectories: 30v2+y2(12y315y280y+120)=c; separatrices: 30v2+y2(12y315y280y+120)=496 and 30v2+y2(12y315y280y+120)=37

9. No equilibria if a<0;0 is unstable if a=0; a is stable and a is unstable if a>0.

10. 0 is a stable equilibrium if a0; a and a are stable and 0 is unstable if a>0.

11. 0 is unstable if a0; a and a are unstable and 0 is stable if a>0.

12. 0 is stable if a0;0 is stable and a and a are unstable if a0.

22. An equilibrium solution ¯y of y+p(y)=0 is unstable if there’s an >0 such that, for every δ>0, there’s a solution of (A) with (y(0)¯y)2+v2(0)<δ, but (y(t)¯y)2+v2(t) for some t>0.


14.16: Section 4.4 Answers is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

Support Center

How can we help?