For which values of the parameter α is the trivial solution…
For which values of the parameter $\alpha$ is the trivial solution of the equation $$y^{(4)}+2y'''+y''+\alpha y'+3y=0$$ 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
$$P(\lambda)=\lambda^4+2\lambda^3+\lambda^2+\alpha\lambda+3,\qquad a_0=1,a_1=2,a_2=1,a_3=\alpha,a_4=3.$$
Step 2 · Hurwitz determinants
$$\Delta_1=2>0,\qquad\Delta_2=2-\alpha,\qquad \Delta_3=2\alpha-12-\alpha^2=-\alpha^2+2\alpha-12=-((\alpha-1)^2+11).$$
Step 3 · Key observation
Since $(\alpha-1)^2+11>0$ for every real $\alpha$, we have $\Delta_3<0$ for all $\alpha\in\mathbb{R}$: the discriminant of $\alpha^2-2\alpha+12$ is $4-48=-44<0$ with positive leading coefficient, so $\alpha^2-2\alpha+12>0$ always.
Step 4 · Routh array
The first column of the Routh array is $1,\;2,\;\tfrac{2-\alpha}{2},\;\tfrac{-\alpha^2+2\alpha-12}{2-\alpha},\;3$. Since the numerator of the fourth entry is always negative, exactly two sign changes occur regardless of the sign of $2-\alpha$ (whether $\alpha<2$ or $\alpha>2$), giving two roots with positive real part in every case. At $\alpha=2$ a zero row appears but the resulting auxiliary-polynomial analysis shows no purely imaginary roots and instability persists.
Step 5 · Conclusion
$$\boxed{ \text{The trivial solution is \textbf{unstable for all values} of }\alpha\in\mathbb{R}. }$$ The Hurwitz determinant $\Delta_3=-((\alpha-1)^2+11)$ is strictly negative for every real $\alpha$, so the Routh--Hurwitz necessary condition $\Delta_3>0$ can never be satisfied: there is no value of $\alpha$ for which the trivial solution is stable.
Summary Table
| Quantity | Value |
|---|---|
| Eigenvalues | — |
| Classification | unstable for all values |
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: