Now solved: Solve the following initial value problems. (Show the details.)

Find a real general solution of the following systems. (Show the details.) \(y_{1}^{\prime}=3 y_{2}\) \(y_{2}^{\prime}=12 y_{1}\) Text Transcription: y_1’ = 3y_2 y_2’ = 12y_1

Find a real general solution of the following systems. (Show the details.) \(y_{1}^{\prime}=5 y_{2}\) \(y_{2}^{\prime}=5 y_{1}\) Text Transcription: y_1’ = 5y_2 y_2’ = 5y_1

Find a real general solution of the following systems. (Show the details.) \(y_{1}^{\prime}=y_{1}+y_{2}\) \(y_{2}^{\prime}=y_{1}+y_{2}\) Text Transcription: y_1’ = y_1 + y_2 y_2’ = y_1 + y_2

Find a real general solution of the following systems. (Show the details.) \(y_{1}^{\prime}=9 y_{1}+13.5 y_{2}\) \(y_{2}^{\prime}=1.5 y_{1}+9 y_{2}\) Text Transcription: y_1’ = 9y_1 + 13.5y_2 y_2’ = 1.5y_1 + 9y_2

Find a real general solution of the following systems. (Show the details.) \(y_{1}^{\prime}=4 y_{2}\) \(y_{2}^{\prime}=-4 y_{1}\) Text Transcription: y_1’ = 4y_2 y_2’ = - 4y_1

Find a real general solution of the following systems. (Show the details.) \(y_{1}^{\prime}=2 y_{1}-2 y_{2}\) \(y_{2}^{\prime}=2 y_{1}+2 y_{2}\) Text Transcription: y_1’ = 2y_1 - 2y_2 y_2’ = 2y_1 + 2y_2

Find a real general solution of the following systems. (Show the details.) \(y_{1}^{\prime}=2 y_{1}+8 y_{2}-4 y_{3}\) \(y_{2}^{\prime}=-4 y_{1}-10 y_{2}+2 y_{3}\) \(y_{3}^{\prime}=-4 y_{1}-4 y_{2}-4 y_{3}\) Text Transcription: y_1’ = 2y_1 + 8y_2 - 4y_3 y_2’ = - 4y_1 - 10y_2 + 2y_3 y_3’ = - 4y_1 - 4y_2 - 4y_3

Find a real general solution of the following systems. (Show the details.) \(y_{1}^{\prime}=8 y_{1}-y_{2}\) \(y_{2}^{\prime}=y_{1}+10 y_{2}\) Text Transcription: y_1’ = 8y_1 - y_2 y_2’ = y_1 + 10y_2

Find a real general solution of the following systems. (Show the details.) \(y_{1}^{\prime}=-y_{1}+y_{2}+0.4 y_{3}\) \(y_{2}^{\prime}=y_{1}-0.1 y_{2}+1.4 y_{3}\) \(y_{3}^{\prime}=0.4 y_{1}+1.4 y_{2}+0.2 y_{3}\) Text Transcription: y_1’ = - y_1 + y_2 + 0.4y_3 y_2’ = y_1 - 0.1y_2 + 1.4y_3 y_3’ = 0.4y_1 + 1.4y_2 + 0.2y_3

Solve the following initial value problems. (Show the details.) \(y_{1}^{\prime}=y_{1}+y_{2}\) \(y_{2}^{\prime}=4 y_{1}+y_{2}\) \(v_{1}(0)=1, y_{2}(0)=6\) Text Transcription: y_1’ = y_1 + y_2 y_2’ = 4y_1 + y_2 v_1(0) = 1, y_2(0) = 6

Solve the following initial value problems. (Show the details.) \(y_{1}^{\prime}=y_{1}+2 y_{2}\) \(y_{2}^{\prime}=\frac{1}{2} y_{1}+y_{2}\) \(y_{1}(0)=16, y_{2}(0)=-2\) Text Transcription: y_1’ = y_1 + 2y_2 y_2’ = 1 / 2 y_1 + y_2 y_1(0) = 16, y_2 (0) = - 2

Solve the following initial value problems. (Show the details.) \(y_{1}^{\prime}=3 v_{1}+2 y_{2}\) \(y_{2}^{\prime}=2 y_{1}+3 y_{2}\) \(y_{1}(0)=7, y_{2}(0)=7\) Text Transcription: y_1 = 3v_1 + 2y_2 y_2 = 2y_1 + 3y_2 y_1 (0) = 7, y_2 (0) = 7

Solve the following initial value problems. (Show the details.) \(y_{1}^{\prime}=\frac{1}{2} y_{1}-2 y_{2}\) \(y_{2}^{\prime}=-\frac{3}{2} v_{1}+y_{2}\) \(y_{1}(0)=0.4, y_{2}(0)=3.8\) Text Transcription: y_1 = 1 / 2 y_1 - 2y_2 y_2 = -3 / 2 v_1 + y_2 y_1 (0) = 0.4, y_2 (0) = 3.8

Solve the following initial value problems. (Show the details.) \(y_{1}^{\prime}=-y_{1}+5 y_{2}\) \(y_{2}^{\prime}=-y_{1}+3 y_{2}\) \(y_{1}(0)=7, y_{2}(0)=2\) Text Transcription: y_1’ = - y_1 + 5y_2 y_2’ = - y_1 + 3y_2 y_1(0) = 7, y_2 (0) = 2

Solve the following initial value problems. (Show the details.) \(y_{1}^{\prime}=2 y_{1}+5 y_{2}\) \(y_{2}^{\prime}=5 y_{1}+12.5 y_{2}\) \(y_{1}(0)=12, v_{2}(0)=1\) Text Transcription: y_1’ = 2y_1 + 5y_2 y_2’ = 5y_1 + 12.5y_2 y_1(0) = 12, v_2(0) = 1

Find a general solution by conversion to a single ODE.The system in Prob. 8.

Find a general solution by conversion to a single ODE. The system in Example 5.

Each of the two tanks contains 400 gal of water. in which initially 100 lb (Tank \(T_{1}\) ) and 40 lb (Tank \(T_{2}\) ) of fertilizer are dissolved. The inflow, circulation, and outflow are shown in Fig. 87. The mixture is kept uniform by stirring. Find the fertilizer contents \(y_{1}(t)\) in \(T_{1}\) and \(y_{2}(t)\) in \(T_{2}\). Text Transcription: . T_1 T_2 y_1(t) y_2(t)

Show that a model for the currents \(I_{1}(t)\) and \(I_{2}(t)\) in Fig. 88 is \(\frac{1}{C} \int I_{1} d t+R\left(I_{1}-I_{2}\right)=0, \quad L I_{2}^{\prime}+R\left(I_{2}-I_{1}\right)=0\). Find a general solution, assuming that \(R=20 \Omega, L=0.5 \mathrm{H}, C=2 \cdot 10^{-4} \mathrm{~F}\). Text Transcription: I_1(t) I_2(t) 1 / C int I_1 dt + R(I_1 - I_2) = 0, LI_2’ + R(I_2 - I_1) = 0 R = 20 Omega, L = 0.5 H, C = 2 cdot 10^-4 F

Phase Portraits. Graph some of the figures in this section, in particular, Fig. 86 on the degenerate node, in which the vector \(y^{(2)}\) depends on t. In each figure highlight a trajectory that satisfies an initial condition of your choice. Text Transcription: y^(2)