91Ó°ÊÓ

Determine the location and type of all critical points by linearization. $$\begin{aligned}&y_{1}^{\prime}=-3 y_{1}+y_{2}-y_{2}^{2}\\\&y_{2}^{\prime}=y_{1}-3 y_{2}\end{aligned}$$

Find a real general solution of the following systems. (Show the details.) $$\begin{array}{l} y_{1}^{\prime}=9 y_{1}+13.5 y_{2} \\ y_{2}^{\prime}=1.5 y_{1}+9 y_{2} \end{array}$$

Find a general solution. (Show the details of your work.) $$\begin{array}{l}y_{1}^{\prime}=2 y_{1}+2 y_{2}+12 \\\y_{2}^{\prime}=5 y_{1}-y_{2}-30\end{array}$$

Determine the location and type of all critical points by linearization. $$\begin{array}{l}y_{1}^{\prime}=-y_{1}+y_{2}-y_{2}^{2} \\\y_{2}^{\prime}=-y_{1}-y_{2}\end{array}$$

Determine the type and stability of the critical point Then find a real general solution and sketch or graph some of the trajoctories in the phase plane. (Show the details of your work.) $$\begin{array}{l}y_{1}^{\prime}=-4 y_{1}+y_{2} \\\y_{2}^{\prime}=y_{1}-4 y_{2}\end{array}$$

Find a real general solution of the following systems. (Show the details.) $$\begin{array}{l} y_{1}^{\prime}=4 y_{2} \\ y_{2}^{\prime}=-4 y_{1} \end{array}$$

Determine the location and type of all critical points by linearization. $$\begin{array}{l}y_{1}^{\prime}=y_{2}-y_{2}^{2} \\\y_{2}^{\prime}=y_{2}-y_{2}^{2}\end{array}$$

Find a general solution. (Show the details of your work.) $$\begin{array}{l}y_{i}^{\prime}=-y_{1}+y_{2}+e^{-2 t} \\\y_{2}^{\prime}=-y_{1}-y_{2}-e^{-2}\end{array}$$

Determine the type and stability of the critical point Then find a real general solution and sketch or graph some of the trajoctories in the phase plane. (Show the details of your work.) $$\begin{array}{l}y_{1}^{\prime}=y_{2}+10 y_{2} \\\y_{2}^{\prime}=7 y_{1}-8 y_{2}\end{array}$$

Determine the type and stability of the critical point Then find a real general solution and sketch or graph some of the trajoctories in the phase plane. (Show the details of your work.) $$\begin{array}{l}y_{1}^{\prime}=-2 y_{2} \\\y_{2}^{\prime}=8 y_{1}\end{array}$$