91Ó°ÊÓ

Apply the eigenvalue method of this section to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. For each problem, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system. $$ x_{1}^{\prime}=x_{1}+2 x_{2}, x_{2}^{\prime}=2 x_{1}+x_{2} $$

Apply the eigenvalue method of this section to find a general solution of the given system. If initial values are given, find also the corresponding particular solution. For each problem, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system. $$ x_{1}^{T}=2 x_{1}+3 x_{2}, \quad x_{2}^{\prime}=2 x_{1}+x_{2} $$

Problems deal with the mass-and-spring system shown in Fig. 5.3.11 with stiffness matrix $$ \mathbf{K}=\left[\begin{array}{cc} -\left(k_{1}+k_{2}\right) & k_{2} \\ k_{2} & -\left(k_{2}+k_{3}\right) \end{array}\right] $$ and with the given mks values for the masses and spring constants. Find the two natural frequencies of the system and describe its two natural modes of oscillation. $$ m_{1}=m_{2}=1 ; k_{1}=1, k_{2}=2, k_{3}=1 $$

Find general solutions of the systems in Problems 1 through 22\. In Problems I through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system. \(\mathbf{x}^{\prime}=\left[\begin{array}{rrr}0 & 0 & 1 \\ -5 & -1 & -5 \\ 4 & 1 & -2\end{array}\right] \mathbf{x}\)

Compute the matrix exponential \(e^{\mathrm{A} t}\) for each system \(\mathrm{x}^{\prime}=\mathrm{Ax}\) given. $$ x_{1}^{\prime}=11 x_{1}-15 x_{2}, x_{2}^{\prime}=6 x_{1}-8 x_{2} $$

Show that the matrix \(A\) is nilpotent and then use this fact to find (as in Example 3) the matrix exponential \(e^{\mathrm{A} l}\). $$ \mathbf{A}=\left[\begin{array}{ll} 1 & -1 \\ 1 & -1 \end{array}\right] $$

Deal with the open three-tank system Fig. 5.2.2. Fresh water flows into tank 1 ; mixed brine flows om tank 1 into tank 2 , from tank 2 into tank 3 , and out of tank all at the given flow rate \(r\) gallons per minute. The initial mounts \(x_{1}(0)=x_{0}(l b), x_{2}(0)=0\), and \(x_{3}(0)=0\) of salt the three tanks are given, as are their volumes \(V_{1}, V_{2}\), and \(\mathrm{V}_{\mathrm{a}}\) (in gallons). First solve for the amounts of salt in the three tanks at time \(t\), then determine the maximal amount of salt that tank 3 ever contains. Finally, construct a figure showing the graphs of \(x_{1}(t), x_{2}(t)\), and \(x_{3}(t) .\) $$ r=60, x_{0}=40, V_{1}=20, V_{2}=12, V_{3}=60 $$