Examine the stability of the trivial solution of the…

Examine the stability of the trivial solution of the equation $$y^{(5)}+y^{(4)}+7y'''+4y''+10y'+3y=0.$$

Step 1 · Reduction to a first-order system

Setting $x_1=y,\ldots,x_5=y^{(4)}$ converts the ODE into a first-order linear system whose characteristic polynomial coincides with that of the ODE. By the theory of constant-coefficient linear systems, the trivial solution is asymptotically stable iff all roots of the characteristic polynomial have strictly negative real parts.

Step 2 · Characteristic polynomial

$$P(\lambda)=\lambda^5+\lambda^4+7\lambda^3+4\lambda^2+10\lambda+3.$$

Step 3 · Rational root test

Testing $\lambda=\pm1,\pm3$ gives $P(-3)=-342$, $P(-1)=-10$, $P(1)=26$, $P(3)=582$ — no rational roots. We apply the Routh--Hurwitz criterion instead of finding roots explicitly.

Step 4 · Routh--Hurwitz criterion

With $a_0=1,a_1=1,a_2=7,a_3=4,a_4=10,a_5=3$, the Hurwitz determinants are: $$\Delta_1=a_1=1,\qquad \Delta_2=\begin{vmatrix}1&4\\1&7\end{vmatrix}=3,\qquad \Delta_3=\begin{vmatrix}1&4&3\\1&7&10\\0&1&4\end{vmatrix}=5,$$ $$\Delta_4=\begin{vmatrix}1&4&3&0\\1&7&10&0\\0&1&4&3\\0&1&7&10\end{vmatrix}=8, \qquad \Delta_5=a_5\Delta_4=3\cdot8=24.$$ All five determinants are strictly positive.

Step 5 · Routh array (verification)

Constructing the Routh array from the coefficients gives first-column entries $1,\;1,\;3,\;\tfrac53,\;\tfrac85,\;3$ — all positive, with no sign changes, confirming the Hurwitz determinant computation.

Step 6 · Summary of Hurwitz determinants

Step 7 · Stability conclusion

Since all roots of $P(\lambda)$ have strictly negative real parts, the trivial solution $y\equiv0$ is Lyapunov stable and, moreover, asymptotically stable: every solution $y(t)\to0$ as $t\to+\infty$.

Step 8 · Conclusion

$$\boxed{ \text{The trivial solution of the equation is \textbf{asymptotically stable}.} }$$ All five Hurwitz determinants are positive ($\Delta_1=1,\Delta_2=3,\Delta_3=5,\Delta_4=8,\Delta_5=24$), confirming that every root of $\lambda^5+\lambda^4+7\lambda^3+4\lambda^2+10\lambda+3=0$ lies in the open left half-plane $\operatorname{Re}(\lambda)<0$, so the trivial solution is asymptotically stable.

Summary Table

Additional Notes

• The Routh–Hurwitz determinants (or, equivalently, the first column of the Routh array) provide a purely algebraic stability test that requires only the coefficients of the characteristic polynomial, with no need to locate its roots.