Using the Laplace transform and showing the details of YOUT work, solve the initial

Using the Laplace transform and showing the details of your work, solve the initial value problem: \(y_{1}^{\prime}=-y_{1}-y_{2}, \quad y_{2}^{\prime}=y_{1}-y_{2}, y_{1}(0)=0, \quad y_{2}(0)=1\) Text Transcription: y’_1=y_1-y_2, y’_2=y_1-y_2, y_1(0)=0, y_2(0)=1

Using the Laplace transform and showing the details of your work, solve the initial value problem: \(y_{1}^{\prime}=5 y_{1}+y_{2}, \quad y_{2}^{\prime}=y_{1}+5 y_{2}, y_{1}(0)=1, \quad y_{2}(0)=-3\) Text Transcription: y’=5y_1+y_2, y’_2=y_1+5y_2, y_1(0)=1, y_2(0)=-3

Using the Laplace transform and showing the details of your work, solve the initial value problem: \(y_{1}^{\prime}=-6 y_{1}+4 y_{2}, \quad y_{2}^{\prime}=-4 y_{1}+4 y_{2}, y_{1}(0)=-2, \quad y_{2}(0)=-7\) Text Transcription: y’_1=-6y_1+4y_2, y’_2=-4y_1+4y_2, y_1(0)=-2, y_2(0)=-7

Using the Laplace transform and showing the details of your work, solve the initial value problem: \(y_{1}^{\prime}+y_{2}=0, \quad y_{1}+y_{2}^{\prime}=2 \cos t, y_{1}(0)=1, \quad y_{2}(0)=0\) Text Transcription: y’_1+y_2=0, y_1+y’_2=2 cos t, y_1(0)=1, y_2(0)=0

Using the Laplace transform and showing the details of your work, solve the initial value problem: \(y_{1}^{\prime}=-4 y_{1}-2 y_{2}+t, \quad y_{2}^{\prime}=3 v_{1}+y_{2}-t, y_{1}(0)=5.75, \quad y_{2}(0)=-6.75\) Text Transcription: y’_1=-4y_1-2y_2 + t, y’_2=3v_1+y_2-t, y_1(0)=5.75, y_2(0)=-6.75

Using the Laplace transform and showing the details of your work, solve the initial value problem: \(y_{1}^{\prime}=4 y_{2}-8 \cos 4 t, \quad y_{2}^{\prime}=-3 y_{1}-9 \sin 4 t, y_{1}(0)=0, \quad y_{2}(0)=3\) Text Transcription: y’_1=4y_2-8 cos 4t, y’_2 = -3y_1 - 9 sin 4t, y_1(0)=0, y_2(0)=3

Using the Laplace transform and showing the details of your work, solve the initial value problem: \(\begin{array}{l} y_{1}^{\prime}=5 y_{1}-4 y_{2}-9 t^{2}+2 t, \\ y_{2}^{\prime}=10 y_{1}-7 y_{2}-17 t^{2}-2 t, \\ y_{1}(0)=2, \quad y_{2}(0)=0 \end{array} \) Text Transcription: y’_1=5y_1-4y_1-4y_2-9t^2+2t, y’_2=10y_1-7y_2-17t^2-2t, y_1(0)=2, y_2(0)=0

Using the Laplace transform and showing the details of your work, solve the initial value problem: \(\begin{array}{ll} y_{1}^{\prime}=6 y_{1}+y_{2}, & y_{2}^{\prime}=9 y_{1}+6 y_{2}, \\ y_{1}(0)=-3, & y_{2}(0)=-3 \end{array} \) Text Transcription: y’_1=6y_1+y_2, y’_2=9y_1+6y_2, y_1(0)=-3, y_2(0)=-3

Using the Laplace transform and showing the details of your work, solve the initial value problem: \(\begin{array}{l} y_{1}^{\prime}=5 y_{1}+5 y_{2}-15 \cos t+27 \sin t, \\ y_{2}^{\prime}=-10 y_{1}-5 y_{2}-150 \sin t, \\ y_{1}(0)=2, \quad y_{2}(0)=2 \end{array} \) Text Transcription: y’_1=5y_1+5y_2-15 cos t + 27 sin t, y’_2=-10y_1-5y_2-150 sin t, y_1(0)=2, y_2(0)=2

Using the Laplace transform and showing the details of your work, solve the initial value problem: \(\begin{array}{l} y_{1}^{\prime}=-2 y_{1}+3 y_{2}, \quad y_{2}^{\prime}=4 y_{1}-y_{2}, \\ y_{1}(0)=4, \quad y_{2}(0)=3 \end{array} \) Text Transcription: y’_1=-2y_1+3y_2, y’_2=4y_1-y_2, y_1(0)=4, y_2(0)=3

Using the Laplace transform and showing the details of your work, solve the initial value problem: \(\begin{array}{l}y_{1}^{\prime}=y_{2}+1-u(t-1), \\y_{2}^{\prime}=-y_{1}+1-u(t-1), \quad y_{1}(0)=0, \\y_{2}(0)=0\end{array}\) Text Transcription: y’_1=y_2+1-u(t-1), y’_2=-y_1+1-u(t-1), y_1(0)=0, y_2(0)=0

Using the Laplace transform and showing the details of your work, solve the initial value problem: \(\begin{array}{ll}y_{1}^{\prime}=2 y_{1}+y_{2}, & y_{2}^{\prime}=4 y_{1}+2 y_{2}+64 t u(t-1), \\ y_{1}(0)=2, & y_{2}(0)=0\end{array}\) Text Transcription: y’_1=2y_1+y_2, y’_2=4y_1+2y_2+64tu(t-1), y_1(0)=2, y_2(0)=0

Using the Laplace transform and showing the details of your work, solve the initial value problem: \(\begin{array}{l} y_{1}^{\prime}=y_{1}+6 u(t-2) e^{4 t}, y_{2}^{\prime}=y_{1}+2 y_{2}, \\y_{1}(0)=0, \quad y_{2}(0)=1\end{array}\) Text Transcription: y’_1=y_1+6u(t-2)e^4t, y’_2=y_1+2y_2, y_1(0)=0, y_2(0)=1

Using the Laplace transform and showing the details of your work, solve the initial value problem: \(\begin{array}{lc}y_{1}^{\prime}=-y_{2}, & y_{2}^{\prime}=-y_{1}+2[1-u(t-2 \pi)] \cos t,\\y_{1}(0)=1, & y_{2}(0)=0\end{array}\) Text Transcription: y’=-y_2, y’_2=-y_1+2[1-u(t-2 pi)] cos t, y_1(0)=1, y_2(0)=0

Using the Laplace transform and showing the details of your work, solve the initial value problem: \(\begin{array}{l}y_{1}^{\prime}=-3 y_{1}+y_{2}+u(t-1) e^{t}, \\y_{2}^{\prime}=-4 y_{1}+2 y_{2}+u(t-1) e^{t}, \\y_{1}(0)=0, \quad y_{2}(0)=3\end{array}\) Text Transcription: y’_1=-3y_1+y_2+u(t-1)e^t, y’_2=-4y_1+2y_2+u(t-1)e^t, y_1(0)=0, y_2(0)=3

Using the Laplace transform and showing the details of your work, solve the initial value problem: \(\begin{array}{l}y_{1}^{\prime \prime}=-2 y_{1}+2 y_{2}, \quad y_{2}^{\prime \prime}=2 y_{1}-5 y_{2}, \\y_{1}(0)=1, \quad y_{1}^{\prime}(0)=0, \quad y_{2}(0)=3, \quad y_{2}^{\prime}(0)=0\end{array}\) Text Transcription: y’’_1=-2y_1+2y_2, y’’_2=2y_1-5y_2, y_1(0)=1, y’_1(0)=0, y_2(0)=3, y’_2(0)=0

Using the Laplace transform and showing the details of your work, solve the initial value problem: \(\begin{array}{l}y_{1}^{\prime \prime}=4 y_{1}+8 y_{2}, \quad y_{2}^{\prime \prime}=5 y_{1}+y_{2}, \\y_{1}(0)=8, \quad y_{1}^{\prime}(0)=-18, \quad y_{2}(0)=5, \\y_{2}^{\prime}(0)=-21\end{array}\) Text Transcription: y’’_1=4y_1+8y_2, y’’_2=5y_1+y_2, y_1(0)=8, y’(0)=-18, y_2(0)=5, y’_2(0)=-21

Using the Laplace transform and showing the details of your work, solve the initial value problem: \(y_{1}^{\prime \prime}+y_{2}=-101 \sin 10 t . \quad y_{2}^{\prime \prime}+y_{1}=101 \sin 10 t, y_{1}(0)=0, \quad y_{1}^{\prime}(0)=6, \quad y_{2}(0)=8, y_{2}^{\prime}(0)=-6\) Text Transcription: y’’_1+y_2=-101 sin 10t, y’’_2+y_1=101 sin 10t, y_1(0)=0, y’_1(0)=6, y_2(0)=8, y’_2(0)=-6

Using the Laplace transform and showing the details of your work, solve the initial value problem: \(\begin{array}{l}y_{1}^{\prime}+y_{2}^{\prime}=2 e^{t}+e^{-t}, \quad y_{2}^{\prime}+y_{3}^{\prime}=2 \sinh t ,\\y_{3}^{\prime}+y_{1}^{\prime}=e^{t} \\y_{1}(0)=0, \quad y_{2}(0)=1, \quad y_{3}(0)=1\end{array}\) Text Transcription: y’_1+y’_2=2e^t+e^-t, y’_2+y’_3=2 sinh t, y’_3+y’_1=e^t y_1(0)=0, y_2(0)=1, y_3(0)=1

Using the Laplace transform and showing the details of your work, solve the initial value problem: \(\begin{array}{l}4 y_{1}^{\prime}+y_{2}^{\prime}-2 y_{3}^{\prime}=0 . \quad-2 y_{1}^{\prime}+y_{3}^{\prime}=1 , \\2 y_{2}^{\prime}-4 y_{3}^{\prime}=-16 t \\y_{1}(0)=2, \quad y_{2}(0)=0, \quad y_{3}(0)=0\end{array}\) Text Transcription: 4y’_1+y’_2-2y’_3=0, -2y’_1+y’_3=1, 2y’_2-4y’_3= -16t y_1(0)=2, y_2(0)=0, y_3(0)=0

TEAM PROJECT. Comparison of Methods for Linear Systems of ODEs. (a) Models. Solve the models in Examples 1 and 2 of Sec. 4.1 by Laplace transforms and compare the amount of work with that in Sec. 4.1. (Show the details of your work.) (b) Homogeneous Systems. Solve the systems (8), (11)-(13) in Sec. 4.3 by Laplace transforms. (Show the details.) (c) Nonhomogeneous System. Solve the system (3) in Sec. 4.6 by Laplace transforms. (Show the details.)

(Forced vibrations of two masses) Solve the model in Example 3 with k = 4 and initial conditions \(y_{1}(0)=1, y_{1}^{\prime}(0)=1, y_{2}(0)=1, y_{2}^{\prime}(0)=-1\) under the assumption that the force 11 sin t is acting on the first body and the force -11 sin t on the second. Graph the two curves on common axes and explain the motion physically. Text Transcription: y_1(0)=1, y’_1(0)=1, y_2(0)=1, y’_2(0)=-1

CAS Experiment. Effect of Initial Conditions. In Prob. 22, Vary the initial conditions systematically, describe and explain the graphs physically. The great variety of curves will surprise you. Are they always periodic? Can you find empirical laws for the changes in terms of continuous changes of those conditions?

(Mixing problem) What will happen in Example 1 if you double all flows (in particular, an increase to 12 gal/min containing 12 lb of salt from the outside), leaving the size of the tanks and the initial conditions as before? First guess, then calculate. Can you relate the new solution to the old one?

(Electrical network) Using Laplace transforms, find the currents \(i_{1}(t)\) and \(i_{2}(t)\) in Fig. 146, where \(v(t)=390 \cos t\) and \(i_{1}(0)=0, i_{2}(0)=0\). How soon will the currents practically reach their steady state? Text Transcription: i_1(t) i_2(t) v(t)=390 cos t i_1(0)=0, i_2(0)=0

(Single cosine wave) Solve Prob. 25 when the EMF (electromotive force) is acting from 0 to \(2 \pi\) only. Can you do this just by looking at Prob. 25, practically without calculation? Text Transcription: 2 pi