91Ó°ÊÓ

In Problems, show that the given system is almost linear with \((0,0)\) as a critical point, and classify this critical point as to type and stability. Use a computer system or graphing calculator to construct a phase plane portrait that illustrates your conclusion. $$ \frac{d x}{d t}=1-e^{x}+2 y, \frac{d y}{d t}=-x-4 \sin y $$

Each of the systems in Problems 11 through 18 has a single critical point \(\left(x_{0}, y_{0}\right) .\) Apply Theorem 2 to classify this critical point as to type and stability. Verify your conclusion by using a computer system or graphing calculator to construct a phase portrait for the given system. $$ \frac{d x}{d t}=x-2 y-8, \quad \frac{d y}{d t}=x+4 y+10 $$

Determine whether the critical point \((0,0)\) is stable, asymptotically stable, or unstable. Use a computer system or graphing calculator to construct a phase portrait and direction field for the given system. Thereby ascertain the stability or instability of each critical point, and identify it visually as a node, a saddle point, a center, or a spiral point. $$ \frac{d x}{d t}=x, \quad \frac{d y}{d t}=3 y $$