In 3740, use diagonalization to solve the given system

In Problems 1–8, use the method of undetermined coefficients to solve the given system. \(\frac{dx}{dt}=2x+3y-7\) \(\frac{dy}{dt}=-x-2y+5\)

In Problems 1–8, use the method of undetermined coefficients to solve the given system. \(\frac{dx}{dt}=5x+9y+2\) \(\frac{dy}{dt}=-x+11y+6\)

In Problems 1–8, use the method of undetermined coefficients to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{ll}1 & 3 \\ 3 & 1\end{array}\right) \mathbf{X}+\left(\begin{array}{c}-2 t^{2} \\ t+5\end{array}\right)\)

In Problems 1–8, use the method of undetermined coefficients to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}1 & -4 \\ 4 & 1\end{array}\right) \mathbf{X}+\left(\begin{array}{rr}4 t+9 e^{6 t} \\ -t+ & e^{6 t}\end{array}\right)\)

In Problems 1–8, use the method of undetermined coefficients to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{ll}4 & \frac{1}{3} \\ 9 & 6\end{array}\right) \mathbf{X}+\left(\begin{array}{r}-3 \\ 10\end{array}\right) e^{t}\)

In Problems 1–8, use the method of undetermined coefficients to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{ll}-1 & 5 \\ -1 & 1\end{array}\right) \mathbf{X}+\left(\begin{array}{c}\sin t \\ -2 \cos t\end{array}\right)\)

In Problems 1–8, use the method of undetermined coefficients to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{lll}1 & 1 & 1 \\ 0 & 2 & 3 \\ 0 & 0 & 5\end{array}\right) \mathbf{X}+\left(\begin{array}{r}1 \\ -1 \\ 2\end{array}\right) e^{4 t}\)

In Problems 1–8, use the method of undetermined coefficients to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{lll}0 & 0 & 5 \\ 0 & 5 & 0 \\ 5 & 0 & 0\end{array}\right) \mathbf{X}+\left(\begin{array}{r}5 \\ -10 \\ 40\end{array}\right)\)

In Problems 9 and 10, solve the given initial-value problem. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}-1 & -2 \\ 3 & 4\end{array}\right) \mathbf{X}+\left(\begin{array}{l}3 \\ 3\end{array}\right), \quad \mathbf{X}(0)=\left(\begin{array}{r}-4 \\ 5\end{array}\right)\)

In Problems 9 and 10, solve the given initial-value problem. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}1 & -1 \\ 1 & 3\end{array}\right) \mathbf{X}+\left(\begin{array}{c}t \\ t+1\end{array}\right), \quad \mathbf{X}(0)=\left(\begin{array}{l}0 \\ 2\end{array}\right)\)

Consider the large mixing tanks shown in FIGURE 10.4.1. Suppose that both tanks A and B initially contain 100 gallons of brine. Liquid is pumped in and out of the tanks as indicated in the figure; the mixture pumped between and out of the tanks is assumed to be well-stirred. (a) Construct a mathematical model in the form of a linear system of first-order differential equations for the number of pounds \(x_{1}(t)\) and \(x_{2}(t)\) of salt in tanks A and B, respectively, at time t. Write the system in matrix form. [Hint: See Section 2.9 and Problem 44, Chapter 2 in Review.] (b) Use the method of undetermined coefficients to solve the linear system in part (a) subject to \(x_{1}(0)=60, x_{2}(0)=10\). (c) What are \(\lim _{t \rightarrow \infty} x_{1}(t)\) and \(\lim _{t \rightarrow \infty} x_{2}(t)\)? Interpret this result. (d) Use a graphing utility or CAS to plot the graphs of \(x_{1}(t)\) and \(x_{2}(t)\) in the same coordinate plane.

(a) The system of differential equations for the currents \(i_{2}(t)\) and \(i_{3}(t)\) in the electrical network shown in FIGURE 10.4.2 is \(\frac{d}{d t}\left(\begin{array}{l} i_{2} \\ i_{3} \end{array}\right)=\left(\begin{array}{cc} -R_{1} / L_{1} & -R_{1} / L_{1} \\ -R_{1} / L_{2} & -\left(R_{1}+R_{2}\right) / L_{2} \end{array}\right)\left(\begin{array}{l} i_{2} \\ i_{3} \end{array}\right)+\left(\begin{array}{l} E / L_{1} \\ E / L_{2} \end{array}\right)\). Use the method of undetermined coefficients to solve the system if \(R_{1}=2 \Omega, R_{2}=3 \Omega, L_{1}=1 \mathrm{~h}, L_{2}=1 \mathrm{~h}\), E = 60 V, \(i_{2}(0)=0\), and \(i_{3}(0)=0\). (b) Determine the current \(i_1(t)\).

In Problems 13–32, use variation of parameters to solve the given system. \(\frac{dx}{dt}=3x-3y+4\) \(\frac{dy}{dt}=2x-2y-1\)

In Problems 13–32, use variation of parameters to solve the given system. \(\frac{dx}{dt}=2x-y\) \(\frac{dy}{dt}=3x-2y+4t\)

In Problems 13–32, use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{cc}3 & -5 \\ \frac{3}{4} & -1\end{array}\right) \mathbf{X}+\left(\begin{array}{r}1 \\ -1\end{array}\right) e^{t / 2}\)

In Problems 13–32, use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}2 & -1 \\ 4 & 2\end{array}\right) \mathbf{X}+\left(\begin{array}{c}\sin 2 t \\ 2 \cos 2 t\end{array}\right) e^{2 t}\)

In Problems 13–32, use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}0 & 2 \\ -1 & 3\end{array}\right) \mathbf{X}+\left(\begin{array}{r}1 \\ -1\end{array}\right) e^{t}\)

In Problems 13–32, use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}0 & 2 \\ -1 & 3\end{array}\right) \mathbf{X}+\left(\begin{array}{c}2 \\ e^{-3 t}\end{array}\right)\)

In Problems 13–32, use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}1 & 8 \\ 1 & -1\end{array}\right) \mathbf{X}+\left(\begin{array}{l}12 \\ 12\end{array}\right) t\)

In Problems 13–32, use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}1 & 8 \\ 1 & -1\end{array}\right) \mathbf{X}+\left(\begin{array}{l}e^{-t} \\ t e^{t}\end{array}\right)\)

In Problems 13–32, use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}3 & 2 \\ -2 & -1\end{array}\right) \mathbf{X}+\left(\begin{array}{c}2 e^{-t} \\ e^{-t}\end{array}\right)\)

In Problems 13–32, use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}3 & 2 \\ -2 & -1\end{array}\right) \mathbf{X}+\left(\begin{array}{l}1 \\ 1\end{array}\right)\)

In Problems 13–32, use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}0 & -1 \\ 1 & 0\end{array}\right) \mathbf{X}+\left(\begin{array}{c}\sec t \\ 0\end{array}\right)\)

In Problems 13–32, use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}1 & -1 \\ 1 & 1\end{array}\right) \mathbf{X}+\left(\begin{array}{l}3 \\ 3\end{array}\right) e^{t}\)

In Problems 13–32, use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}1 & -1 \\ 1 & 1\end{array}\right) \mathbf{X}+\left(\begin{array}{c}\cos t \\ \sin t\end{array}\right) e^{t}\)

In Problems 13–32, use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{ll}2 & -2 \\ 8 & -6\end{array}\right) \mathbf{X}+\left(\begin{array}{l}1 \\ 3\end{array}\right) \frac{e^{-2 t}}{t}\)

In Problems 13–32, use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right) \mathbf{X}+\left(\begin{array}{c}0 \\ \sec t \tan t\end{array}\right)\)

In Problems 13–32, use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right) \mathbf{X}+\left(\begin{array}{c}1 \\ \cot t\end{array}\right)\)

In Problems 13–32, use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}1 & 2 \\ -\frac{1}{2} & 1\end{array}\right) \mathbf{X}+\left(\begin{array}{c}\csc t \\ \sec t\end{array}\right) e^{t}\)

In Problems 13–32, use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{ll}1 & -2 \\ 1 & -1\end{array}\right) \mathbf{X}+\left(\begin{array}{c}\tan t \\ 1\end{array}\right)\)

In Problems 13–32, use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{lll}1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 3\end{array}\right) \mathbf{X}+\left(\begin{array}{c}e^{t} \\ e^{2 t} \\ t e^{3 t}\end{array}\right)\)

In Problems 13–32, use variation of parameters to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rrr}3 & -1 & -1 \\ 1 & 1 & -1 \\ 1 & -1 & 1\end{array}\right) \mathbf{X}+\left(\begin{array}{c}0 \\ t \\ 2 e^{t}\end{array}\right)\)

In Problems 33 and 34, use (14) to solve the given initial-value problem. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}3 & -1 \\ -1 & 3\end{array}\right) \mathbf{X}+\left(\begin{array}{l}4 e^{2 t} \\ 4 e^{4 t}\end{array}\right), \quad \mathbf{X}(0)=\left(\begin{array}{l}1 \\ 1\end{array}\right)\)

In Problems 33 and 34, use (14) to solve the given initial-value problem. \(\mathbf{X}^{\prime}=\left(\begin{array}{ll}1 & -1 \\ 1 & -1\end{array}\right) \mathbf{X}+\left(\begin{array}{c}1 / t \\ 1 / t\end{array}\right), \quad \mathbf{X}(1)=\left(\begin{array}{r}2 \\ -1\end{array}\right)\)

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 \(i_{2}(0)=0\).

Solving a nonhomogeneous linear system \(\mathbf{X}^\prime=\mathbf{AX+F}(t)\) by variation of parameters when A is a 3 X 3 (or larger) matrix is almost an impossible task to do by hand. Consider the system \(\mathbf{X}^{\prime}=\left(\begin{array}{rrrr} 2 & -2 & 2 & 1 \\ -1 & 3 & 0 & 3 \\ 0 & 0 & 4 & -2 \\ 0 & 0 & 2 & -1 \end{array}\right) \mathbf{X}+\left(\begin{array}{c} t e^{t} \\ e^{-t} \\ e^{2 t} \\ 1 \end{array}\right)\). (a) Use a CAS or linear algebra software to find the eigenvalues and eigenvectors of the coefficient matrix. (b) Form a fundamental matrix \(\boldsymbol{\Phi}(t)\) and use the computer to find \(\boldsymbol{\Phi}^{-1}(t)\). (c) Use the computer to carry out the computations of \(\boldsymbol{\Phi}^{-1}(t) \mathbf{F}(t), \int \boldsymbol{\Phi}^{-1}(t) \mathbf{F}(t) d t, \boldsymbol{\Phi}(t) \int \boldsymbol{\Phi}^{-1}(t) \mathbf{F}(t) d t, \boldsymbol{\Phi}(t) \mathbf{C}\), and \(\boldsymbol{\Phi}(t) \mathbf{C}+\int \boldsymbol{\Phi}^{-1}(t) \mathbf{F}(t) d t\), where C is a column matrix of constants \(c_{1}, c_{2}, c_{3}\), and \(c_{4}\). (d) Rewrite the computer output for the general solution of the system in the form \(\mathbf{X}=\mathbf{X}_{c}+\mathbf{X}_{p}\), where \(\mathbf{X}_{c}=c_{1} \mathbf{X}_{1}+c_{2} \mathbf{X}_{2}+c_{3} \mathbf{X}_{3}+c_{4} \mathbf{X}_{4}\).

In Problems 37-40, use diagonalization to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{rr}5 & -2 \\ 21 & -8\end{array}\right) \mathbf{X}+\left(\begin{array}{l}6 \\ 4\end{array}\right)\)

In Problems 37-40, use diagonalization to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{ll}1 & 3 \\ 2 & 2\end{array}\right) \mathbf{X}+\left(\begin{array}{c}e^{t} \\ e^{t}\end{array}\right)\)

In Problems 37-40, use diagonalization to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{ll}5 & 5 \\ 5 & 5\end{array}\right) \mathbf{X}+\left(\begin{array}{c}2 t \\ 8\end{array}\right)\)

In Problems 37-40, use diagonalization to solve the given system. \(\mathbf{X}^{\prime}=\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right) \mathbf{X}+\left(\begin{array}{c}4 \\ 8 e^{-2 t}\end{array}\right)\)