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

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

Step 1 · Coefficient matrix

$$A=\begin{pmatrix}-3&2\\1&1\end{pmatrix}.$$

Step 2 · Characteristic equation and eigenvalues

$$\det(A-\lambda I)=0\implies\lambda^2+2\lambda-5=0.$$ Discriminant $\Delta=24>0$, so $$\lambda_1=-1+\sqrt{6}\approx1.449,\qquad\lambda_2=-1-\sqrt{6}\approx-3.449.$$

Step 3 · Classification of the rest point

The eigenvalues are real, distinct, and of opposite sign. The rest point $(0,0)$ is a saddle point: it is unstable.

Step 4 · Eigenvectors and separatrices

For $\lambda_1=-1+\sqrt6$: eigenvector $(\sqrt6-2,1)^\top$, giving the unstable separatrix tangent to $y=\tfrac{\sqrt6+2}{2}x$ (departs the origin as $t\to+\infty$). For $\lambda_2=-1-\sqrt6$: eigenvector $(-(2+\sqrt6),1)^\top$, giving the stable separatrix (attracted to the origin as $t\to+\infty$).

Step 5 · General solution

$$x(t)=C_1(\sqrt6-2)e^{\lambda_1 t}-C_2(2+\sqrt6)e^{\lambda_2 t},\qquad y(t)=C_1e^{\lambda_1 t}+C_2e^{\lambda_2 t}.$$

Step 6 · Behaviour of integral curves

Eliminating $t$ gives the homogeneous ODE $dy/dx=(x+y)/(-3x+2y)$, whose integral curves form the characteristic hyperbolic family of a saddle, asymptotic to the two eigendirections found above.

Step 7 · Stability analysis

Since $\lambda_1=-1+\sqrt6>0$, the rest point is unstable: there exist trajectories (all those not on the stable separatrix) that diverge from the origin.

Step 8 · Conclusion

$$\boxed{ (0,0)\text{ is a \textbf{saddle point} (unstable).} }$$ The eigenvalues $\lambda_1=-1+\sqrt6\approx1.449>0$ and $\lambda_2=-1-\sqrt6\approx-3.449<0$ are real with opposite signs. The unstable separatrix is tangent to $\mathbf v^{(1)}=(\sqrt6-2,1)^\top$ and the stable separatrix to $\mathbf v^{(2)}=(-(2+\sqrt6),1)^\top$ at the origin. All other trajectories exhibit the characteristic hyperbolic divergence of a saddle.

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.