Examine the character of the rest point (0,0) of the system…

Examine the character of the rest point $(0,0)$ of the system $$\dot{x}=3x-5y,\qquad\dot{y}=2x-3y.$$

Step 1 · Coefficient matrix

$$A=\begin{pmatrix}3&-5\\2&-3\end{pmatrix}.$$

Step 2 · Characteristic equation and eigenvalues

$$\det(A-\lambda I)=0\implies(3-\lambda)(-3-\lambda)+10=0 \implies\lambda^2+1=0\implies\lambda_{1,2}=\pm i.$$

Step 3 · Classification of the rest point

The eigenvalues are purely imaginary. Since the system is linear (not merely linearised), the rest point is a centre exactly, not subject to the indeterminacy that arises for nonlinear systems.

Step 4 · Direction of rotation

Evaluating $\dot y$ at $(1,0)$ gives $\dot y=2>0$, so the trajectories rotate counter-clockwise.

Step 5 · General real solution

With eigenvector $\mathbf v=(3,2)^\top+i(1,0)^\top$ for $\lambda=i$: $$x(t)=(3C_1+C_2)\cos t+(-C_1+3C_2)\sin t,\qquad y(t)=2C_1\cos t+2C_2\sin t.$$

Step 6 · First integral and family of trajectories

Seeking a quadratic invariant $F=ax^2+bxy+cy^2$ with $\dot F\equiv0$ along trajectories gives (up to scale) $a=2$, $b=-6$, $c=5$, so the family of integral curves is $$2x^2-6xy+5y^2=K,\quad K>0,$$ a family of concentric ellipses centred at the origin (positive definite since the discriminant $36-40=-4<0$ with positive leading coefficient).

Step 7 · Behaviour of trajectories

Since $\operatorname{Re}(\lambda_{1,2})=0$, every trajectory is a closed ellipse; trajectories neither approach nor recede from the origin. The angular frequency is $\omega=1$, giving period $T=2\pi$.

Step 8 · Stability analysis

The rest point is stable in the sense of Lyapunov (trajectories remain on a fixed ellipse $2x^2-6xy+5y^2=K$ close to the origin for small $K$) but not asymptotically stable, since no trajectory converges to the origin.

Step 9 · Conclusion

$$\boxed{ (0,0)\text{ is a \textbf{centre} (stable, not asymptotically stable).}\quad \lambda_{1,2}=\pm i. }$$ All trajectories are closed ellipses $2x^2-6xy+5y^2=K$ ($K>0$) traversed counter-clockwise with period $T=2\pi$. The origin is Lyapunov stable but not asymptotically stable.

Summary Table

Additional Notes

• The classification follows from the standard theory of planar linear autonomous systems, based on the sign and nature of the eigenvalues of the coefficient matrix.