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

Examine the character of the rest point $(0,0)$ of the system $$\dot{x}=-3x+\tfrac{7}{2}y,\qquad\dot{y}=-x+y.$$

Step 1 · Coefficient matrix

$$A=\begin{pmatrix}-3&\tfrac{7}{2}\\-1&1\end{pmatrix}.$$

Step 2 · Characteristic equation and eigenvalues

$$\det(A-\lambda I)=0\implies\lambda^2+2\lambda+\tfrac12=0 \implies2\lambda^2+4\lambda+1=0.$$ Discriminant $\Delta=8>0$, so $$\lambda_1=\frac{-2+\sqrt{2}}{2}\approx-0.293,\qquad \lambda_2=\frac{-2-\sqrt{2}}{2}\approx-1.707.$$

Step 3 · Classification of the rest point

Both eigenvalues are real, distinct, and negative. The rest point $(0,0)$ is a stable node: all trajectories approach the origin as $t\to+\infty$.

Step 4 · Eigenvectors

For $\lambda_1$: eigenvector $(4-\sqrt2,\,2)^\top$, giving the slower nodal axis $y=\tfrac{4+\sqrt2}{7}x$. For $\lambda_2$: eigenvector $(4+\sqrt2,\,2)^\top$, giving the faster nodal axis $y=\tfrac{4-\sqrt2}{7}x$.

Step 5 · General solution

$$x(t)=C_1(4-\sqrt2)e^{\lambda_1 t}+C_2(4+\sqrt2)e^{\lambda_2 t},\qquad y(t)=2C_1e^{\lambda_1 t}+2C_2e^{\lambda_2 t}.$$

Step 6 · Behaviour of trajectories near the origin

Since both eigenvalues are negative, all trajectories converge to the origin as $t\to+\infty$. For generic initial conditions ($C_1\neq0$) the approach is tangential to the slower eigendirection $y=\tfrac{4+\sqrt2}{7}x$ (since $e^{\lambda_1 t}$ decays more slowly than $e^{\lambda_2 t}$); only trajectories exactly on the faster eigendirection approach along $y=\tfrac{4-\sqrt2}{7}x$.

Step 7 · Stability analysis

Since both eigenvalues satisfy $\operatorname{Re}(\lambda_{1,2})<0$, the rest point is stable and asymptotically stable.

Step 8 · Conclusion

$$\boxed{ (0,0)\text{ is a \textbf{stable node} (asymptotically stable).} }$$ The eigenvalues $\lambda_1\approx-0.293$ and $\lambda_2\approx-1.707$ are both real, distinct, and negative. All trajectories approach the origin tangentially to the slower eigendirection $\mathbf v^{(1)}=(4-\sqrt2,2)^\top$, except those lying exactly on the faster eigendirection $\mathbf v^{(2)}=(4+\sqrt2,2)^\top$.

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.