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

Examine the character of the rest point $(0,0)$ of the system $$\frac{dx}{dt}=x-y,\qquad\frac{dy}{dt}=2x+3y.$$

Step 1 · Coefficient matrix

The system is linear and homogeneous with coefficient matrix $$A=\begin{pmatrix}1&-1\\2&3\end{pmatrix}.$$

Step 2 · Characteristic equation and eigenvalues

$$\det(A-\lambda I)=0\implies(1-\lambda)(3-\lambda)+2=0 \implies\lambda^2-4\lambda+5=0.$$ Discriminant $\Delta=-4<0$, so $$\lambda_{1,2}=2\pm i.$$

Step 3 · Classification of the rest point

The eigenvalues are complex with positive real part: $\operatorname{Re}(\lambda_{1,2})=2>0$. The rest point $(0,0)$ is an unstable focus. All trajectories spiral away from the origin as $t\to+\infty$.

Step 4 · Direction of rotation

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

Step 5 · Eigenvector and general real solution

For $\lambda_1=2+i$: eigenvector $\mathbf v^{(1)}=(1,-1-i)^\top =(1,-1)^\top+i(0,-1)^\top$. With $\alpha=2$, $\beta=1$, $\mathbf p=(1,-1)^\top$, $\mathbf q=(0,-1)^\top$: $$x(t)=e^{2t}(C_1\cos t+C_2\sin t),$$ $$y(t)=e^{2t}\bigl[(-C_1+C_2)\cos t+(-C_1-C_2)\sin t\bigr].$$

Step 6 · Behaviour of trajectories

The distance from the origin satisfies $r(t)\propto e^{2t}\to+\infty$ as $t\to+\infty$, confirming outward spiraling. As $t\to-\infty$, $r(t)\to0$ and trajectories approach the origin while completing infinitely many counter-clockwise revolutions with angular frequency $\omega=1$ (period $T=2\pi$).

Step 7 · Stability analysis

Since $\operatorname{Re}(\lambda_{1,2})=2>0$, the rest point is unstable: it is not Lyapunov stable, since trajectories starting arbitrarily close to the origin eventually leave any prescribed neighbourhood.

Step 8 · Conclusion

$$\boxed{ (0,0)\text{ is an \textbf{unstable focus}.}\quad \lambda_{1,2}=2\pm i,\quad \operatorname{Re}(\lambda)=2>0. }$$ All trajectories spiral counter-clockwise away from the origin with angular frequency $\omega=1$ and exponential growth rate $\alpha=2$.

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.