

1. $$\overline{y}=0$$ 0 is a stable equilibrium; trajectories are $$v^{2}+\frac{y^{2}}{4}=c$$

2. $$\overline{y}=0$$ 0 is an unstable equilibrium; trajectories are $$v^{2}+\frac{2y^{3}}{3}=c$$

3. $$\overline{y}=0$$ 0 is a stable equilibrium; trajectories are $$v^{2}+\frac{2|y|^{3}}{3}=c$$

4. $$\overline{y}=0$$ 0 is a stable equilibrium; trajectories are $$v^{2}-e^{-y}(y+1)=c$$

5. equilibria: $$0$$ (stable) and $$−2, 2$$ (unstable); trajectories: $$2v^{2} − y^{4} + 8y^{2} = c$$; separatrix: $$2v^{2} − y^{4} + 8y^{2} = 16$$

6. equilibria: $$0$$ (unstable) and $$−2, 2$$ (stable); trajectories: $$2v^{2} + y^{4} − 8y^{2} = c$$; separatrix: $$2v^{2} + y^{4} − 8y^{2} =0$$

7. equilibria: $$0, −2, 2$$ (stable), $$−1, 1$$ (unstable); trajectories: $$6v^{2} + y^{2}(2y^{4} − 15y^{2} + 24) = c$$; separatrix: $$6v^{2} + y^{2} (2y^{4} − 15y^{2} + 24) = 11$$

8. equilibria: $$0, 2$$ (stable) and $$−2, 1$$ (unstable); trajectories: $$30v^{2} + y^{2}(12y^{3} − 15y^{2} − 80y + 120) = c$$; separatrices: $$30v^{2} + y^{2} (12y^{3} − 15y^{2} − 80y + 120) = 496$$ and $$30v^{2} + y^{2} (12y^{3} − 15y^{2} − 80y + 120) = 37$$

9. No equilibria if $$a < 0; 0$$ is unstable if $$a = 0$$; $$\sqrt{a}$$ is stable and $$−\sqrt{a}$$ is unstable if $$a > 0$$.

10. $$0$$ is a stable equilibrium if $$a ≤ 0$$; $$−\sqrt{a}$$ and $$\sqrt{a}$$ are stable and $$0$$ is unstable if $$a > 0$$.

11. $$0$$ is unstable if $$a ≤ 0$$; $$−\sqrt{a}$$ and $$\sqrt{a}$$ are unstable and $$0$$ is stable if $$a > 0$$.

12. $$0$$ is stable if $$a ≤ 0; 0$$ is stable and $$−\sqrt{a}$$ and $$\sqrt{a}$$ are unstable if $$a ≤ 0$$.

22. An equilibrium solution $$\overline{y}$$ 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 $$\sqrt{(y(0)-\overline{y})^{2}+v^{2}(0)}<\delta$$, but $$\sqrt{(y(t)-\overline{y})^{2}+v^{2}(t)}\geq €$$ for some $$t>0$$.