Solved: The system of differential equations for the currents i1(t) and i2(t) in the

Chapter 10, Problem 35

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The system of differential equations for the currents \(i_{1}(t)\) and \(i_{2}(t)\) in the electrical network shown in FIGURE 10.4.3 is

\(\frac{d}{d t}\left(\begin{array}{l} i_{1} \\ i_{2} \end{array}\right)=\left(\begin{array}{cc} -\left(R_{1}+R_{2}\right) / L_{2} & R_{2} / L_{2} \\ R_{2} / L_{1} & -R_{2} / L_{1} \end{array}\right)\left(\begin{array}{l} i_{1} \\ i_{2} \end{array}\right)+\left(\begin{array}{c} E / L_{2} \\ 0 \end{array}\right)\).

Use variation of parameters to solve the system if \(R_{1}=8 \Omega, R_{2}=3 \Omega, L_{1}=1 \mathrm{~h}, L_{2}=1\) h, E(t) = 100 sin t V, \(i_{1}(0)=0\), and \(i_{2}(0)=0\).