For which values of the parameters α and β is the trivial…
For which values of the parameters $\alpha$ and $\beta$ is the trivial solution of the equation $$y'''+\alpha y''+2y'+\beta y=0$$ stable?
Explanation of Method
The Routh–Hurwitz criterion is applied with the coefficients of the characteristic polynomial left as functions of the free parameter(s). Requiring every coefficient and every Hurwitz determinant to be strictly positive translates into a system of inequalities on the parameter(s), carving out the region of asymptotic stability; the boundary of that region, where a determinant vanishes, is checked separately and typically corresponds to simple purely imaginary roots (marginal stability).
Step 1 · Characteristic polynomial
$$P(\lambda)=\lambda^3+\alpha\lambda^2+2\lambda+\beta,\qquad a_0=1,a_1=\alpha,a_2=2,a_3=\beta.$$
Step 2 · Routh--Hurwitz conditions
Necessary: $\alpha>0$, $\beta>0$. Hurwitz condition: $$\Delta_2=2\alpha-\beta>0\implies\beta<2\alpha.$$ Asymptotic stability region: $\alpha>0,\;\beta>0,\;\beta<2\alpha$.
Step 3 · Boundary $\beta=2\alpha$ ($\alpha>0$)
$$P(\lambda)=(\lambda+\alpha)(\lambda^2+2),$$ roots $\lambda_1=-\alpha<0$, $\lambda_{2,3}=\pm i\sqrt2$ (simple): stable, not asymptotically.
Step 4 · Boundary $\beta=0$ ($\alpha>0$)
$$P(\lambda)=\lambda(\lambda^2+\alpha\lambda+2),$$ $\lambda=0$ simple, remaining roots have negative real part for $\alpha>0$: stable, not asymptotically.
Step 5 · Boundary $\alpha=0$
$P(\lambda)=\lambda^3+2\lambda+\beta$: necessary condition fails, unstable.
Step 6 · Summary table
| Condition | $\Delta_2$ | Stability |
|---|---|---|
| $\alpha>0,\beta>0,\beta<2\alpha$ | $>0$ | Asymptotically stable |
| $\alpha>0,\beta>0,\beta=2\alpha$ | $=0$ | Stable (not asympt.) |
| $\alpha>0,\beta=0$ | $2\alpha>0$ | Stable (not asympt.) |
| otherwise | $\leq0$ or fails | Unstable |
Step 7 · Conclusion
$$\boxed{ \text{Asymptotically stable:}\quad\alpha>0,\quad\beta>0,\quad\beta<2\alpha. }$$ $$\boxed{ \text{Stable (not asymptotically):}\quad\alpha>0,\quad(\beta=2\alpha\text{ or }\beta=0). }$$ The stability region is the open wedge $\{(\alpha,\beta):\alpha>0,0<\beta<2\alpha\}$ in the $(\alpha,\beta)$ plane, bounded by the marginal-stability lines $\beta=2\alpha$ and $\beta=0$.
Summary Table
| Condition | $\Delta_2$ | Stability |
|---|---|---|
| $\alpha>0,\beta>0,\beta<2\alpha$ | $>0$ | Asymptotically stable |
| $\alpha>0,\beta>0,\beta=2\alpha$ | $=0$ | Stable (not asympt.) |
| $\alpha>0,\beta=0$ | $2\alpha>0$ | Stable (not asympt.) |
| otherwise | $\leq0$ or fails | Unstable |
Additional Notes
- The boundary curves or points where a Hurwitz determinant vanishes mark the transition between asymptotic stability and instability; on such a boundary the characteristic polynomial typically factors to reveal simple purely imaginary roots, giving stability without asymptotic stability exactly on that boundary.
Result
The final classification is: