(Network) Find the currents in Fig. 100. where R1 = 0.5 fl, R2 = 0.7 fl, Ll = 0.4 H, L2

State some applications that can be modeled by systems of ODEs.

What is population dynamics? Give examples.

How can you transform an ODE into a system of ODEs?

What are qualitative methods for systems? Why are they important?

What is the phase plane? The phase plane method? The phase portrait of a system of ODEs?

What is a critical point of a system of ODEs? How did we classify these points?

What are eigenvalues? What role did they play in this chapter?

What does stability mean in general? In connection with critical points?

What does linearization of a system mean? Give an example.

What is a limit cycle? When may it occur in mechanics?

Find a general solution. Determine the kind and stability of the critical point. (Show the details of your work.) \(y_{1}^{\prime}=4 y_{2}\) \(y_{2}^{\prime}=16 y_{1} Text Transcription: y_1’ = 4y_2 y_2’ = 16y_1

Find a general solution. Determine the kind and stability of the critical point. (Show the details of your work.) \(y_{1}^{\prime}=9 y_{1}\) \(y_{2}^{\prime}=y_{2}\) Text Transcription: y_1’ = 9y_1 y_2’ = y_2

Find a general solution. Determine the kind and stability of the critical point. (Show the details of your work.) \(y_{1}^{\prime}=y_{2}\) \(y_{2}^{\prime}=6 y_{1}-5 y_{2}\) Text Transcription: y_1’ = y_2 y_2’ = 6y_1 - 5y_2

Find a general solution. Determine the kind and stability of the critical point. (Show the details of your work.) \(y_{1}^{\prime}=3 y_{1}-3 y_{2}\) \(y_{2}^{\prime}=3 y_{1}+3 y_{2}\) Text Transcription: y_1’ = 3y_1 - 3y_2 y_2’ = 3y_1 + 3y_2

Find a general solution. Determine the kind and stability of the critical point. (Show the details of your work.) \(y_{1}^{\prime} =1.5 y_{1}-6 y_{2}\) \(y_{2}^{\prime} =-4.5 y_{1}+3 y_{2}\) Text Transcription: y_1’ = 1.5y_1 - 6y_2 y_2’ = -4.5y_1 + 3y_2

Find a general solution. Determine the kind and stability of the critical point. (Show the details of your work.) \(y_{1}^{\prime} =-3 y_{1}-2 y_{2}\) \(y_{2}^{\prime}=-2 y_{1}-3 y_{2}\) Text Transcription: y_1’ = -3y_1 - 2y_2 y_2’ = -2y_1 - 3y_2

Find a general solution. Determine the kind and stability of the critical point. (Show the details of your work.) \(y_{1}^{\prime}=3 y_{1}+2 y_{2} \(y_{2}^{\prime}=2 y_{1}+3 y_{2}\) Text Transcription: y_1’ = 3y_1 + 2y_2 y_2’ = 2y_1 + 3y_2

Find a general solution. Determine the kind and stability of the critical point. (Show the details of your work.) \(y_{1}^{\prime}=3 y_{1}+5 y_{2}\) \(y_{2}^{\prime}=-5 y_{1}-3 y_{2}\) Text Transcription: y_1’ = 3y_1 + 5y_2 y_2’ = -5y_1 - 3y_2

Find a general solution. Determine the kind and stability of the critical point. (Show the details of your work.) \(y_{1}^{\prime}=-y_{1}+2 y_{2}\) \(y_{2}^{\prime}=-2 y_{1}-y_{2}\) Text Transcription: y_1’ = -y_1 + 2y_2 y_2’ = -2y_1 - y_2

Find a general solution. (Show the details.) \(y_{1}^{\prime}=3 y_{2}+6 t\) \(y_{2}^{\prime}=12 y_{1}+1\) Text Transcription: y_1’ = 3y_2 + 6t y_2’ = 12y_1 + 1

Find a general solution. (Show the details.) \(y_{1}^{\prime}=y_{1}+2 y_{2}+e^{2 t}\) \(y_{2}^{\prime}=-y_{2}+1.5 e^{-2 t}\) Text Transcription: y_1’ = y_1 + 2y_2 + e^2t y_2’ = -y_2 + 1.5 e^-2t

Find a general solution. (Show the details.) \(y_{1}^{\prime}=y_{1}+y_{2}+\sin t\) \(y_{2}^{\prime}=4 y_{1}+y_{2}\) Text Transcription: y_1’ = y_1 + y_2 + sin t y_2’ = 4y_1 + y_2

Find a general solution. (Show the details.) \(y_{1}^{\prime}=4 y_{1}+3 y_{2}+2\) \(y_{2}^{\prime}=-6 y_{1}-5 y_{2}+4 e^{-t}\) Text Transcription: y_1’ = 4y_1 + 3y_2 + 2 y_2’ = -6y_1 - 5y_2 + 4e^-t

Find a general solution. (Show the details.) \(y_{1}^{\prime}=y_{1}-2 y_{2}-\sin t\) \(y_{2}^{\prime}=3 y_{1}-4 y_{2}-\cos t\) Text Transcription: y_1’ = y_1 - 2y_2 - sin t y_2’ = 3y_1 - 4y_2 - cos t

Find a general solution. (Show the details.) \(y_{1}^{\prime}=y_{1}+2 y_{2}+t^{2}\) \(y_{2}^{\prime}=2 y_{1}+y_{2}-t^{2}\) Text Transcription: y_1’ = y_1 + 2y_2 + t^2 y_2’ = 2y_1 + y_2 - t^2

Tank \(T_{1}\) in Fig. 99 contains initially 200 gal of water in which 160 lb of salt are dissolved. Tank \(T_{2}\) contains initially 100 gal of pure water. Liquid is pumped through the system as indicated. and the mixtures are kept uniform by stirring. Find the amounts of salt \(y_{1}(t)\) and \(y_{2}(t)\) in \(T_{1}\) and \(T_{2}\), respectively. Text Transcription: T_1 T_2 y_1(t) y_2(t)

What kind of critical point does y’ = Ay have if A has the eigenvalues -6 and 1?

(Network) Find the currents in Fig. 100 , where \(R_{1}=0.5 \Omega, R_{2}=0.7 \Omega, L_{1}=0.4 \mathrm{H}, L_{2}=0.5 \mathrm{H}\), \(E=1 \mathrm{kV}=1000 \mathrm{~V}\), and \(I_{1}(0)=0, I_{2}(0)=0\). Text Transcription: R_1 = 0.5 Omega, R_2 = 0.7 Omega, L_1 = 0.4 H, L_2 = 0.5 H E = 1 kV = 1000 V I_1(0) = 0, I_2 (0) = 0

(Network) Find the currents in Fig. 101 when \(R=10 \Omega\), L = 1.25 H, C = 0.002 F, and \(I_{1}(0)=I_{2}(0)=3 \mathrm{~A}\). Text Transcription: R = 10 Omega I_1(0) = I_2(0) = 3 A

Determine the location and kind of all critical points of the given nonlinear system by linearization. \(y_{1}^{\prime}=y_{2}\) \(y_{2}^{\prime}=4 y_{1}-y_{1}^{ }{3}\) Text Transcription: y_1’ = y_2 y_2’ = 4y_1 - y_1^3

Determine the location and kind of all critical points of the given nonlinear system by linearization. \(y_{1}^{\prime}=-9 y_{2}\) \(y_{2}^{\prime}=\sin y_{1}\) Text Transcription: y_1’ = -9y_2 y_2’ = sin y_1

Determine the location and kind of all critical points of the given nonlinear system by linearization. \(y_{1}^{\prime}=\cos y_{2}\) \(y_{2}^{\prime}=3 y_{1}\) Text Transcription: y_1’ = cos y_2 y_2’ = 3y_1