Examine the stability of the trivial solution of the…
Examine the stability of the trivial solution of the equation $$y''+10y=0.$$
Explanation of Method
The characteristic roots turn out to be purely imaginary — the borderline “centre” case for a linear constant-coefficient equation. Multiplying the equation by the appropriate factor produces a conserved quantity (the mechanical energy of the corresponding oscillator), which serves directly as a Lyapunov function: it is positive definite and constant along every trajectory, giving Lyapunov stability without asymptotic stability.
Step 1 · Characteristic polynomial
The equation is a second-order linear ODE with constant coefficients. The characteristic polynomial is: $$P(\lambda)=\lambda^2+10.$$
Step 2 · Roots of the characteristic polynomial
$$\lambda^2+10=0\implies\lambda^2=-10\implies\lambda_{1,2}=\pm\sqrt{10}\,i.$$ The roots are purely imaginary.
Step 3 · Routh--Hurwitz analysis
For $P(\lambda)=\lambda^2+a_1\lambda+a_2$ with $a_1=0$, $a_2=10$, the condition $a_1>0$ fails, signalling a critical (purely imaginary) case rather than instability.
Step 4 · General solution
$$y(t)=C_1\cos\sqrt{10}\,t+C_2\sin\sqrt{10}\,t.$$
Step 5 · First integral — conservation of energy
Multiplying by $y'$ gives the conserved quantity $$E(t)=\frac{y'(t)^2}{2}+5y(t)^2=\text{const}.$$
Step 6 · Verification of Lyapunov stability
Take $V(y,y')=\tfrac{y'^2}{2}+5y^2$, positive definite with $\dot V=0$. A standard $\varepsilon$-$\delta$ argument using the bounds $\tfrac12(y^2+y'^2)\leq V\leq5(y^2+y'^2)$ shows the trivial solution is Lyapunov stable.
Step 7 · Non-asymptotic stability
Since $V$ is constant and positive along any non-trivial solution, $y(t)\not\to0$: the origin is a centre, not asymptotically stable.
Step 8 · Conclusion
$$\boxed{ \text{The trivial solution is \textbf{stable} (in the sense of Lyapunov) but \textbf{not asymptotically stable}.} }$$
Summary Table
| Quantity | Value |
|---|---|
| Eigenvalues | — |
| Classification | stable / not asymptotically stable |
Additional Notes
- A conserved energy-like quantity, once shown to be a positive definite Lyapunov function with zero derivative along trajectories, immediately yields Lyapunov stability of a centre — without ever needing to solve the equation or invoke the linearisation theorem.
Result
The final classification is: