Chapter 10 Linear Systems of Differential Equations

\(\begin{array}{rcl} Q_1'&=&\dst{2-{1\over10}Q_1+{1\over25}Q_2}\\[6pt] Q_2'&=&\dst{6+{3\over50}Q_1-{1\over20}Q_2}. \end{array}\)

\(\begin{array}{rcl} Q_1'&=&\dst{12-{5\over100+2t}Q_1+{1\over100+3t}Q_2}\\[6pt] Q_2'&=&\dst{5+{1\over50+t}Q_1-{4\over100+3t}Q_2}. \end{array}\)

\(m_1y_1''=-(c_1+c_2)y_1'+c_2y_2'-(k_1+k_2)y_1+k_2y_2+F_1\)

   \(m_2y_2''=(c_2-c_3)y_1'-(c_2+c_3)y_2'+c_3y_3'+(k_2-k_3)y_1 -(k_2+k_3)y_2 +k_3y_3+F_2\)

   \(m_3y_3''=c_3y_1'+c_3y_2'-c_3y_3'+k_3y_1+k_3y_2-k_3y_3+F_3\)

\(x''=-\dst{{\alpha\over m}x'+{gR^2x \over(x^2+y^2+z^2)^{3/2}}}\)  \(y''=-\dst{{\alpha\over m}y'+{gR^2y \over\left(x^2+y^2+z^2\right)^{3/2}}}\)

   \(z''=-\dst{{\alpha\over m}z'+{gR^2z \over\left(x^2+y^2+z^2\right)^{3/2}}}\)

   (e)\( \begin{array}{ccl}x_1'&=&x_2\\x_2'&=&f(t,x_1,y_1)\\ y_1'&=&y_2\\y_2'&=&g(t,x_1,y_1)\end{array}\)

(a) \({\bf y}'=\twobytwo2442{\bf y}\)     (b) \({\bf y}'=\twobytwo{-2}{-2}{-5}1{\bf y}\)

   (c) \({\bf y}'=\twobytwo{-4}{-10}37{\bf y}\)  (d) \({\bf y}'=\twobytwo2112{\bf y}\)

(a) \({\bf y}'=\threebythree{-1}2301600{-2}{\bf y}\)   (b) \({\bf y}'=\threebythree022202220{\bf y}\)

   (c) \({\bf y}'=\threebythree{-1}222{-1}222{-1}{\bf y}\)   (d) \({\bf y}'=\threebythree3{-1}{-1}{-2}324{-1}{-2}{\bf y}\)

(a) \({\bf y}'=\twobytwo11{-2}4{\bf y},\; {\bf y}(0)=\twocol10\)   (b) \({\bf y}'=\twobytwo53{-1}1{\bf y},\; {\bf y}(0)=\twocol9{-5}\)

(a) \({\bf y}'=\threebythree644{-7}{-2}{-1}743{\bf y},\; {\bf y}(0) =\threecol3{-6}4\)

   (b) \({\bf y}'=\threebythree877{-5}{-6}{-9}57{10}{\bf y},\; {\bf y}(0) =\threecol2{-4}3\)

(a) \({\bf y}'=\twobytwo{-3}2{-5}3{\bf }+\twocol{3-2t}{6-3t}\)   (b) \({\bf y}'=\twobytwo31{-1}1{\bf y}+\left[\begin{array}{c}-5e^t\\e^t\end{array}\right]\)

(a)  \(\dst{d\over dt}Y^2=Y'Y+YY'\)

     (b)  \(\dst{d\over dt}Y^{n}= Y'Y^{n-1}+YY'Y^{n-2}+Y^2Y'Y^{n-3}+\cdots+Y^{n-1}Y'= \sum_{r=0}^{n-1} Y^rY'Y^{n-r-1}\)

\(B=(P'+PA)P^{-1}\).

\(\dst{{\bf y}'=\left[\begin{array}{cc}0&1\\[3pt]-\dst{P_2(x)\over P_0(x) }&-\dst{P_1(x)\over P_0(x)}\end{array}\right]{\bf y}}\)  

\({\bf y}'=\left[\begin{array}{cccc} 0&1&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\0&0&\cdots&1\\[3pt] -\dst{P_n(x)\over P_0(x)}&-\dst{P_{n-1}(x)\over P_0(x)}&\cdots& -\dst{P_1(x)\over P_0(x)}\end{array}\right]{\bf y}\)

(b)  \({\bf y}=\dst{\left[\begin{array}{c}3e^{6t}-6e^{-2t}\\3e^{6t}+6e^{-2t} \end{array}\right]}\)   (c)  \({\bf y}= \dst{{1\over2}\left[\begin{array}{cc}e^{6t}+e^{-2t}&e^{6t}-e^{-2t} \\e^{6t}-e^{-2t}&e^{6t}+e^{-2t}\end{array}\right]}{\bf k}\)

  (b) \({\bf y}=\dst{\left[\begin{array}{c}6e^{-4t}+4e^{3t}\\6e^{-4t}-10e^{3t} \end{array}\right]}\)   (c)  \({\bf y}= \dst{{1\over7}\left[\begin{array}{cc}5e^{-4t}+2e^{3t}&2e^{-4t}-2e^{3t} \\5e^{-4t}-5e^{3t}&2e^{-4t}+5e^{3t}\end{array}\right]}{\bf k}\)

(b) \({\bf y}=\dst{\left[\begin{array}{c}-15e^{2t}-4e^t\\9e^{2t}+2e^t \end{array}\right]}\)    (c) \({\bf y}=\dst{\left[\begin{array}{cc}-5e^{2t}+6e^t&-10e^{2t}+10e^t \\3e^{2t}-3e^t&6e^{2t}-5e^t\end{array}\right]}{\bf k}\)

(b) \({\bf y}=\dst{\left[\begin{array}{c}5e^{3t}-3e^t\\5e^{3t}+3e^t \end{array}\right]}\)       (c) \(\dst{\bf y}={{1\over2}\left[\begin{array}{cc}e^{3t}+e^t&e^{3t}-e^t \\e^{3t}-e^t&e^{3t}+e^t\end{array}\right]}{\bf k}\)

(b) \({\bf y }=\dst{\left[\begin{array}{c}e^{2t}-2e^{3t}+3e^{-t}\\2e^{3t}-9e^{-t}\\ e^{2t}-2e^{3t}+21e^{-t}\end{array}\right]}\) (c) \({\bf y}= \dst{{1\over6}\left[\begin{array}{ccc}4e^{2t}+3e^{3t}-e^{-t}&6e^{2t}-6e^{3t} &2e^{2t}-3e^{3t}+e^{-t} \\-3e^{3t}+3e^{-t}&6e^{3t}&3e^{3t}-3e^{-t}\\ 4e^{2t}+3e^{3t}-7e^{-t}&6e^{2t}-6e^{3t}&2e^{2t}-3e^{3t}+7e^{-t} \end{array}\right]}{\bf k}\)

(b) \({\bf y }=\dst{{1\over3}\left[\begin{array}{c}-e^{-2t}+e^{4t}\\-10e^{-2t}+e^{4t}\\ 11e^{-2t}+e^{4t}\end{array}\right]}\)   (c) \({\bf y}= \dst{{1\over3}\left[\begin{array}{ccc}2e^{-2t}+e^{4t}&-e^{-2t}+e^{4t} &-e^{-2t}+e^{4t} \\-e^{-2t}+e^{4t}&2e^{-2t}+e^{4t}&-e^{-2t}+e^{4t}\\ -e^{-2t}+e^{4t}&-e^{-2t}+e^{4t}&2e^{-2t}+e^{4t} \end{array}\right]}{\bf k}\)

(b) \({\bf y }=\dst{\left[\begin{array}{c}3e^t+3e^{-t}-e^{-2t}\\3e^t+2e^{-2t}\\ -e^{-2t}\end{array}\right]}\)   (c) \({\bf y}=\dst{\left[\begin{array}{ccc}e^{-t}&e^t-e^{-t} &2e^t-3e^{-t}+e^{-2t}\\0&e^t&2e^t-2e^{-2t}\\ 0&0&e^{-2t}\end{array}\right]}{\bf k}\)

\(YZ^{-1}\) and \(ZY^{-1}\)

\({\bf y}=\dst{c_1\left[\begin{array}{r}1\\1\end{array} \right]e^{3t}+c_2\left[\begin{array}{r}1\\-1\end{array}\right] e^{-t}}\)  

\({\bf y}=\dst{c_1\left[\begin{array}{r}1\\1\end{array} \right]e^{-t/2}+c_2\left[\begin{array}{r}-1\\1\end{array}\right]e^{-2t}}\)

\({\bf y}=\dst{c_1\left[\begin{array}{r}-3\\1\end{array} \right]e^{-t}+c_2\left[\begin{array}{r}-1\\2\end{array} \right]e^{-2t}}\)

\({\bf y}=\dst{c_1\left[\begin{array}{r}2\\1\end{array} \right]e^{-3t}+c_2\left[\begin{array}{r}-2\\1\end{array} \right]e^t}\)

\({\bf y}=\dst{c_1\left[\begin{array}{r}1\\1 \end{array}\right]e^{-2t}+c_1\left[\begin{array}{r}-4\\1\end{array} \right]e^{3t}}\)  

\({\bf y}=\dst{c_1\left[\begin{array}{r}3\\2\end{array} \right]e^{2t}+c_2\left[\begin{array}{r}1\\1\end{array}\right]e^t}\)

\({\bf y}=\dst{c_1\left[\begin{array}{r}-3\\1 \end{array} \right]e^{-5t}+c_2\left[\begin{array}{r}-1\\1\end{array} \right]e^{-3t}}\)

\({\bf y}=\dst{c_1\left[\begin{array}{r}1\\2\\1 \end{array}\right]e^{-3t}+c_2\left[\begin{array}{r}-1\\-4\\1 \end{array} \right]e^{-t}+c_3\left[\begin{array}{r}-1\\-1\\1\end{array} \right]e^{2t}}\)

\({\bf y}=\dst{c_1\left[\begin{array}{r}2\\1\\2 \end{array}\right]e^{-16t}+c_2\left[\begin{array}{r}-1\\2\\ 0\end{array} \right]e^{2t}+c_3\left[\begin{array}{r}-1\\0\\1 \end{array}\right]e^{2t}}\)

\({\bf y}=\dst{c_1\left[\begin{array}{r}-2\\-4\\3 \end{array}\right]e^t+c_2\left[\begin{array}{r}-1\\1\\ 0\end{array}\right] e^{-2t}+c_3\left[\begin{array}{r}-7\\-5\\4\end{array}\right] e^{2t}}\)

\({\bf y}=\dst{c_1\left[\begin{array}{r}-1\\-1\\1 \end{array}\right]e^{-2t}+c_2\left[\begin{array}{r}-1\\-2\\1 \end{array} \right]e^{-3t}+c_3\left[\begin{array}{r}-2\\-6\\3\end{array} \right]e^{-5t}}\)

\({\bf y}=\dst{c_1\left[\begin{array}{r}11\\\phantom{1}7 \\\phantom{1}1\end{array}\right]e^{3t}+c_2\left[\begin{array}{r}1\\2 \\1\end{array}\right]e^{-2t}+c_3\left[\begin{array}{r}1\\1\\1 \end{array}\right]e^{-t}}\)

\({\bf y}=\dst{c_1\left[\begin{array}{r}4\\-1\\1\end{array} \right]e^{-4t}+c_2\left[\begin{array}{r}-1\\-1\\1 \end{array}\right]e^{6t}+c_3\left[\begin{array}{r}-1\\0\\1 \end{array}\right]e^{4t}}\)

\({\bf y}=\dst{c_1\left[\begin{array}{r}1\\1\\5 \end{array}\right]e^{-5t}+c_2\left[\begin{array}{r}-1\\0\\ 1\end{array} \right]e^{5t}+c_3\left[\begin{array}{r}1\\1\\0\end{array} \right]e^{5t}}\)

\({\bf y}=\dst{c_1\left[\begin{array}{r}1\\ -1\\2 \end{array}\right]+c_2\left[\begin{array}{r}-1\\0\\3\end{array} \right]e^{6t}+c_3\left[\begin{array}{r}1\\3\\0 \end{array}\right]e^{6t}}\)

\({\bf y}=\dst{-\twocol26e^{5t}+\twocol42e^{-5t}}\)

\({\bf y}=\dst{\twocol2{-4}e^{t/2}+\twocol{-2}1e^{t}}\)

\({\bf y}=\dst{\twocol77e^{9t}-\twocol24e^{-3t}}\)

\({\bf y}=\dst{\twocol39e^{5t}-\twocol42e^{-5t}}\)

\({\bf y}=\dst{\threecol550e^{t/2}+\threecol001e^{t/2}+ \threecol{-1}20e^{-t/2}}\)  

\({\bf y}=\dst{\threecol333e^t+\threecol{-2}{-2}2e^{-t}}\)

\({\bf y}=\dst{\threecol2{-2}2e^t-\threecol303e^{-2t}+\threecol110e^{3t}}\)

\({\bf y}=\dst{-\threecol121e^t+\threecol424e^{-t}+\threecol110e^{2t}}\)

\({\bf y}=\dst{ \threecol{-2}{-2}2e^{2t}-\threecol030e^{-2t}+ \threecol4{12}4e^{4t}}\)

\({\bf y}=\dst{ \threecol{-1}{-1}1e^{-6t}+\threecol2{-2}2e^{2t}+ \threecol7{-7}{-7}e^{4t}}\)

\({\bf y}=\dst{ \threecol144e^{-t}+\threecol66{-2}e^{2t}}\)

\({\bf y}=\dst{ \threecol4{-2}2+\threecol3{-9}6e^{4t}+\threecol{-1}1{-1}e^{2t}}\)

Half lines of \(L_1:y_2=y_1\) and \(L_2:y_2=-y_1\) are trajectories other trajectories

are asymptotically tangent to \(L_1\) as \(t\to-\infty\) and asymptotically tangent to \(L_2\) as \(t\to\infty\).

Half lines of \(L_1:y_2=-2y_1\) and \(L_2:y_2=-y_1/3\) are trajectories

other trajectories are asymptotically parallel to \(L_1\) as \(t\to-\infty\) and asymptotically tangent to \(L_2\) as \(t\to\infty\).

Half lines of \(L_1:y_2=y_1/3\) and \(L_2:y_2=-y_1\) are trajectories other trajectories

are asymptotically tangent to \(L_1\) as \(t\to-\infty\) and asymptotically parallel to \(L_2\) as \(t\to\infty\).

Half lines of \(L_1:y_2=y_1/2\) and \(L_2:y_2=-y_1\) are trajectories other trajectories

are asymptotically tangent to \(L_1\) as \(t\to-\infty\) and asymptotically tangent to \(L_2\) as \(t\to\infty\).

Half lines of \(L_1:y_2=-y_1/4\) and \(L_2:y_2=-y_1\) are trajectories other trajectories

are asymptotically tangent to \(L_1\) as \(t\to-\infty\) and asymptotically parallel to \(L_2\) as \(t\to\infty\).

Half lines of \(L_1:y_2=-y_1\) and \(L_2:y_2=3y_1\) are trajectories other trajectories

are asymptotically parallel to \(L_1\) as \(t\to-\infty\) and asymptotically tangent to \(L_2\) as \(t\to\infty\).

Points on \(L_2:y_2=y_1\) are trajectories of constant solutions. The trajectories

of nonconstant solutions are half-lines on either side of \(L_1\), parallel to \(\left[\begin{array}{r} 1\\{-1} \end{array}\right]\), traversed toward \(L_1\).

Points on \(L_1:y_2=-y_1/3\) are trajectories of constant solutions. The trajectories

of nonconstant solutions are half-lines on either side of \(L_1\), parallel to \(\left[\begin{array}{r} -1\\2 \end{array}\right]\), traversed away from \(L_1\).

Points on \(L_1:y_2=y_1/3\) are trajectories of constant solutions. The trajectories

of nonconstant solutions are half-lines on either side of \(L_1\), parallel to \(\left[\begin{array}{r} 1\\{-1} \end{array}\right]\), \(\dst{\twocol{\phantom-}1{-1}}\), traversed away from \(L_1\).

Points on \(L_1:y_2=y_1/2\) are trajectories of constant solutions. The trajectories

of nonconstant solutions are half-lines on either side of \(L_1\), parallel to \(\left[\begin{array}{r} 1\\-1 \end{array}\right]\), \(L_1\).

Points on \(L_2:y_2=-y_1\) are trajectories of constant solutions. The trajectories

of nonconstant solutions are half-lines on either side of \(L_2\), parallel to \(\left[\begin{array}{r} -4\\1 \end{array}\right]\), traversed toward \(L_1\).

Points on \(L_1:y_2=3y_1\) are trajectories of constant solutions. The trajectories

of nonconstant solutions are half-lines on either side of \(L_1\), parallel to \(\left[\begin{array}{r} 1\\{-1} \end{array}\right]\), traversed away from \(L_1\).

\({\bf y}=\dst{c_1\twocol21e^{5t} +c_2\left(\twocol{-1}{\phantom{-}0}e^{5t}+\twocol21te^{5t}\right)}\).

\({\bf y}=\dst{ c_1\twocol11e^{-t}+c_2\left(\twocol10e^{-t}+\twocol11te^{-t}\right)}\)

\({\bf y}=\dst{ c_1\twocol{-2}{\phantom{-}1}e^{-9t}+c_2\left(\twocol{-1}{\phantom{-}0}e^{-9t} +\twocol{-2}{\phantom{-}1}te^{-9t}\right)}\)

\({\bf y}=\dst{ c_1\twocol{-1}{\phantom{-}1}e^{2t}+c_2\left(\twocol{-1}{\phantom{-}0} e^{2t}+\twocol{-1}{\phantom{-}1}te^{2t}\right)}\)

\(c_{1} \left[\begin{array}{r} -2\\1 \end{array}\right] +c_{2}\left( \left[\begin{array}{r} -1\\0 \end{array}\right] \frac{e^{-2t}}{3}+ \left[\begin{array}{r} -2\\1 \end{array}\right] te^{-2t}\right)\)

\({\bf y}=\dst{c_1\twocol32e^{-4t}+ c_2\left(\twocol{-1}{\phantom{-}0}{e^{-4t}\over2}+\twocol32te^{-4t}\right)}\)

\({\bf y}=\dst{c_1\twocol43e^{-t}+ c_2\left(\twocol{-1}{\phantom{-}0}{e^{-t}\over3}+\twocol43te^{-t}\right)}\)

\({\bf y}=\dst{c_1\threecol{-1}{-1}2+c_2\threecol112e^{4t}+ c_3\left(\threecol010{e^{4t}\over2}+\threecol112te^{4t}\right)}\)

\({\bf y}=\dst{c_1\threecol{-1}11e^t+c_2\threecol1{-1}1e^{-t}+c_3\left (\threecol030e^{-t}+\threecol1{-1}1te^{-t}\right)}\).

\({\bf y}=\dst{c_1\threecol011e^{2t}+c_2\threecol101e^{-2t}+c_3\left (\threecol110{e^{-2t}\over2}+\threecol101te^{-2t}\right)}\)

\({\bf y}=\dst{ c_1\threecol{-2}{-3}1e^{2t}+c_2\threecol0{-1}1e^{4t}+c_3\left (\threecol100{e^{4t}\over2}+\threecol0{-1}1te^{4t}\right)}\)

\({\bf y}=\dst{ c_1\threecol{-1}{-1}1e^{-2t}+c_2\threecol111e^{4t}+c_3\left (\threecol100{e^{4t}\over2}+\threecol111te^{4t}\right)}\).

\({\bf y}=\dst{\twocol62e^{-7t}-\twocol84te^{-7t}}\)

\({\bf y}=\dst{\twocol58e^{3t}-\twocol{12}{16}te^{3t}}\)

\({\bf y}=\dst{\twocol23e^{-5t}-\twocol84te^{-5t}}\)

\({\bf y}=\dst{\twocol31e^{5t}-\twocol{12}6te^{5t}}\)

\({\bf y}=\dst{\twocol02e^{-4t}+\twocol66te^{-4t}}\)

\({\bf y}=\dst{\threecol48{-6}e^t+\threecol2{-3}{-1}e^{-2t}+ \threecol{-1}10te^{-2t}}\)

\({\bf y}=\dst{\threecol336e^{2t}-\threecol956+\threecol220t}\)

\({\bf y}=\dst{-\threecol202e^{-3t}+\threecol{-4}91e^t- \threecol044te^t}\)

\({\bf y}=\dst{\threecol{-2}22e^{4t}+\threecol0{-1}1e^{2t}+ \threecol3{-3}3te^{2t}}\)

\({\bf y}=\dst{-\threecol110e^{-4t}+\threecol{-3}2{-3}e^{8t}+ \threecol80{-8}te^{8t}}\)

\({\bf y}=\dst{\threecol363e^{4t}-\threecol341+\threecol844t}\)

\({\bf y}=\dst{c_1\threecol011e^{6t}+ c_2\left(\threecol{-1}10{e^{6t}\over4}+\threecol011te^{6t}\right)}\)

\({\bf y}=\dst{c_1\threecol{-1}11e^{3t}+ c_2\left(\threecol100{e^{3t}\over2}+\threecol{-1}11te^{3t}\right)}\)

\({\bf y}=\dst{c_1\threecol0{-1}1e^{-2t}+ c_2\left(\threecol{{-1}}10e^{-2t}+\threecol0{-1}1te^{-2t}\right)}\)

\({\bf y}=\dst{c_1\threecol011e^{2t}+ c_2\left(\threecol110{e^{2t}\over2}+\threecol011te^{2t}\right)}\)

\({\bf y}=\dst{c_1\threecol{-2}12e^{-6t}+ c_2\left(-\threecol610{e^{-6t}\over6}+\threecol{-2}12te^{-6t}\right)}\)

\({\bf y}=\dst{c_1\threecol{-4}01e^{-3t}+c_2\threecol610e^{-3t}+ c_3\left(\threecol100e^{-3t}+\threecol211te^{-3t}\right)}\)

\({\bf y}=\dst{c_1\threecol{-1}01e^{-3t}+c_2\threecol010e^{-3t}+c_3\left (\threecol100e^{-3t}+\threecol{-1}{-1}1te^{-3t}\right)}\)

\({\bf y}=\dst{ c_1\threecol201e^{-t}+c_2\threecol{-3}20e^{-t}+c_3\left (\threecol100{e^{-t}\over2}+\threecol{-1}21te^{-t}\right)}\)

\({\bf y}=\dst{ c_1\threecol{-1}10e^{-2t}+c_2\threecol001e^{-2t}+c_3\left (\threecol{-1}00e^{-2t}+\threecol1{-1}1te^{-2t}\right)}\)

\({\bf y}=\dst{c_1e^{2t}\ctwocol{3\cos t+\sin t}{5\cos t} +c_2e^{2t}\ctwocol{3\sin t-\cos t}{5\sin t}}\).

\({\bf y}=\dst{c_1e^{-t}\ctwocol{5\cos2t+\sin2t}{13\cos2t} +c_2e^{-t}\ctwocol{5\sin2t-\cos2t}{13\sin2t}}\).

\({\bf y}=\dst{c_1e^{3t}\ctwocol{\cos2t+\sin2t}{2\cos2t} +c_2e^{3t}\ctwocol{\sin2t-\cos2t}{2\sin2t}}\).

\({\bf y}=\dst{c_1e^{2t}\ctwocol{\cos3t-\sin3t}{\cos3t} +c_2e^{2t}\ctwocol{\sin3t+\cos3t}{\sin3t}}\).

\(\dst{{\bf y}=c_1\threecol{-1}{-1}2e^{-2t}+c_2e^{4t}\cthreecol{\cos2t-\sin 2t}{\cos2t+\sin 2t}{2\cos2t}+c_3e^{4t}\cthreecol {\sin2t+\cos2t}{\sin2t-\cos2t}{2\sin2t}}\).

\(\dst{{\bf y}=c_1\threecol{-1}{-1}1e^{-t}+c_2e^{-2t}\cthreecol{\cos2t-\sin 2t}{-\cos2t-\sin 2t}{2\cos2t}+c_3e^{-2t}\cthreecol {\sin2t+\cos2t}{-\sin2t+\cos2t}{2\sin2t}}\)

\(\dst{{\bf y}=c_1\threecol111e^{2t}+c_2e^t\threecol{-\sin t}{\sin t}{\cos t}+c_3e^t\threecol {\cos t}{-\cos t}{\sin t}}\)

\(\dst{{\bf y}=c_1\threecol{-1}11e^t+c_2e^{-t}\cthreecol{-\sin 2t-\cos2t}{2\cos2t}{2\cos2t}+c_3e^{-t}\cthreecol {\cos2t-\sin2t}{2\sin2t}{2\sin2t}}\)

\(\dst{{\bf y}=c_1e^{3t}\left[\begin{array}{c}\cos6t-3\sin6t \\5\cos6t\end{array}\right]+ c_2e^{3t}\left[\begin{array}{c} \sin6t+3\cos6t\\5\sin6t\end{array}\right]}\)

\(\dst{{\bf y}=c_1e^{2t}\left[\begin{array}{c}\cos t-3\sin t \\2\cos t\end{array}\right]+ c_2e^{2t}\left[\begin{array}{c} \sin t+3\cos t\\ 2\sin t\end{array}\right]}\)

\(\dst{{\bf y}=c_1e^{2t}\left[\begin{array}{c}3\sin3t-\cos3t \\ 5\cos3t\end{array}\right]+ c_2e^{2t}\left[\begin{array}{c} -3\cos3t-\sin3t\\ 5\sin3t\end{array}\right]}\)

\(\dst{{\bf y}=c_1e^{2t}\left[\begin{array}{c}\sin4t-8\cos4t \\ 5\cos4t\end{array}\right]+ c_2e^{2t}\left[\begin{array}{c} -\cos4t-8\sin4t\\ 5\sin4t\end{array}\right]}\)

\(\dst{{\bf y}=c_1\left[\begin{array}{r}-1\\1\\1 \end{array}\right]e^{-2t}+c_2e^t\left[\begin{array}{r}\sin t\\-\cos t\\\cos t \end{array}\right]+c_3e^t\left[\begin{array}{r}-\cos t \\-\sin t\\\sin t \end{array}\right]}\)

\(\dst{{\bf y}=c_1\left[\begin{array}{c} 2\\ 2\\1 \end{array}\right]e^{-2t}+ c_2e^{2t}\left[\begin{array}{c}-\cos3t-\sin3t\\-\sin3t\\ \cos3t\end{array}\right]+c_3e^{2t}\left[\begin{array}{c} -\sin3t+\cos3t\\\cos3t\\\sin3t\end{array}\right]}\)

\(\dst{{\bf y}=c_1\left[\begin{array}{r}1\\2\\1 \end{array}\right]e^{3t}+c_2e^{6t}\left[\begin{array}{r}-\sin3t \\\sin3t\\\cos3t \end{array}\right]+ c_3e^{6t}\left[\begin{array}{r}\cos3t \\-\cos3t\\\sin3t\end{array}\right]}\)

\(\dst{{\bf y}=c_1\left[\begin{array}{c} 1\\1\\1 \end{array}\right]e^t+ c_2e^t\left[\begin{array}{c} 2\cos t-2\sin t \\\cos t-\sin t\\ 2\cos t\end{array}\right]+ c_3e^t\left[\begin{array}{ccc} 2\sin t+2\cos t\\\cos t+\sin t\\ 2\sin t\end{array}\right]}\)

\(\dst{{\bf y}=e^t\left[\begin{array}{c} 5\cos3t+\sin3t\\2\cos3t+3\sin3t\end{array}\right]}\)

\(\dst{{\bf y}=e^{4t}\left[\begin{array}{c}5\cos6t+5\sin6t\\\cos6t-3\sin6t \end{array}\right]}\)

\(\dst{{\bf y}=e^t\left[\begin{array}{c}17\cos3t-\sin3t\\7\cos3t+3\sin3t \end{array}\right]}\)

\({\bf y}=\dst{e^{t/2}\left[\begin{array}{c}\cos(t/2)+ \sin(t/2)\\-\cos(t/2)+2\sin(t/2) \end{array}\right]}\)

\(\dst{{\bf y}=\threecol1{-1}2e^t +e^{4t}\left[\begin{array}{c}3\cos t+\sin t\\\cos t-3\sin t\\4\cos t-2\sin t\end{array}\right]}\)

\(\dst{{\bf y}=\threecol442e^{8t} +e^{2t}\left[\begin{array}{c}4\cos2t+8\sin2t\\-6\sin2t+2\cos2t\\3\cos2t+\sin2t \end{array}\right]}\)

\(\dst{{\bf y}=\threecol033e^{-4t} +e^{4t}\left[\begin{array}{c}15\cos6t +10\sin6t\\14\cos6t-8\sin6t\\7\cos6t-4\sin6 t\end{array}\right]}\)

\(\dst{{\bf y}=\threecol6{-3}3e^{8t} +\left[\begin{array}{c}10\cos4t-4\sin4t\\17\cos4t-\sin4t\\3\cos4t-7\sin4t \end{array}\right]}\)

\({\bf U}=\dst{{1\over\sqrt2}\twocol{-1}1}\), \({\bf V}=\dst{{1\over\sqrt2}\twocol11}\) 

\({\bf U}\approx\dst{\twocol{.5257}{.8507}}\), \({\bf V}\approx\dst{\twocol{-.8507}{\phantom{-}.5257}}\)

\({\bf U}\approx\dst{\twocol{.8507}{.5257}}\),

\({\bf V}\approx\dst{\twocol{-.5257}{\phantom{-}.8507}}\) 

\({\bf U}\approx\dst{\twocol{-.9732}{\phantom{-}.2298}}\), \({\bf V}\approx\dst{\twocol{.2298}{.9732}}\)

\({\bf U}\approx\dst{\twocol{\phantom{-}.5257}{.8507}}\), \({\bf V}\approx\dst{\twocol{-.8507}{.5257}}\) 

\({\bf U}\approx\dst{\twocol{-.5257}{\phantom{-}.8507}}\), \({\bf V}\approx\dst{\twocol{.8507}{.5257}}\)

\({\bf U}\approx\dst{\twocol{-.8817}{\phantom{-}.4719}}\), \({\bf V}\approx\dst{\twocol{.4719}{.8817}}\)  

\({\bf U}\approx\dst{\twocol{.8817}{.4719}}\), \({\bf V}\approx\dst{\twocol{-.4719}{\phantom{-}.8817}}\)

\({\bf U}=\dst{\twocol01}\), \({\bf V}=\dst{\twocol{-1}0}\)           

\({\bf U}=\dst{\twocol01}\), \({\bf V}=\dst{\twocol10}\)

\({\bf U}=\dst{{1\over\sqrt2}\twocol11}\), \({\bf V}=\dst{{1\over\sqrt2}\twocol{-1}{\phantom{-}1}}\)   

\({\bf U}\approx\dst{\twocol{.5257}{.8507}}\), \({\bf V}\approx\dst{\twocol{-.8507}{\phantom{-}.5257}}\)

\(\dst{\left[\begin{array}{c}5e^{4t}+e^{-3t}(2+8t)\\ -e^{4t}-e^{-3t}(1-4t)\end{array}\right]}\)

\(\dst{\left[\begin{array}{c}13e^{3t}+3e^{-3t}\\ -e^{3t}-11e^{-3t}\end{array}\right]}\)

\(\dst{{1\over9}\left[\begin{array}{c}7-6t\\ -11+3t\end{array}\right]}\)

\(\dst{\left[\begin{array}{c}5-3e^t\\ -6+5e^t\end{array}\right]}\)

\(\dst{\left[\begin{array}{c}e^{-5t}(3+6t)+e^{-3t}(3-2t)\\ -e^{-5t}(3+2t)-e^{-3t}(1-2t)\end{array}\right]}\)

\(\dst{\left[\begin{array}{c}t\\0 \end{array}\right]}\)

\(\dst{-{1\over6}\left[\begin{array}{c}2-6t\\7+6t\\1-12t \end{array}\right]}\)

\(\dst{-{1\over6}\left[\begin{array}{c}3e^t+4\\6e^t-4\\10 \end{array}\right]}\)

\(\dst{1\over18}{\left[\begin{array}{c}e^t(1+12t)-e^{-5t}(1+6t)\\-2e^t(1-6t)- e^{-5t}(1-12t)\\e^t(1+12t)-e^{-5t}(1+6t) \end{array}\right]}\)

\(\dst{{1\over3}\left[\begin{array}{r}2e^t\\e^t\\2e^t \end{array}\right]}\)

\(\dst{\left[\begin{array}{c}t\sin t\\0 \end{array}\right]}\)

\(\dst{-\left[\begin{array}{c}t^2\\2t \end{array}\right]}\)

\(\dst{(t-1)\left(\ln|t-1|+t\right)\left[\begin{array}{r}1\\-1 \end{array}\right]}\)

\(\dst{{1\over9}\left[\begin{array}{c}5e^{2t}-e^{-3t}\\e^{3t}-5e^{-2t} \end{array}\right]}\)

\(\dst{{1\over4t}\left[\begin{array}{c}2t^3\ln|t|+t^3(t+2)\\ 2\ln|t|+3t-2 \end{array}\right]}\)

\(\dst{{1\over2}\left[\begin{array}{c}te^{-t}(t+2)+(t^3-2)\\ te^t(t-2)+(t^3+2) \end{array}\right]}\)

\(\dst{-\left[\begin{array}{c}t\\t\\t \end{array}\right]}\)

\(\dst{1\over4}\left[\begin{array}{cccc}-3e^t\\ 1\\e^{-t}\end{array}\right]\)

\(\left[\begin{array}{cccc}2t^2+t\\t\\-t\end{array}\right]\)

\(\dst{{e^t\over4t}\left[\begin{array}{c}2t+1\\2t-1\\2t+1 \end{array}\right]}\)

(a) \(\dst{{\bf y}'= \left[\begin{array}{cccc} 0&1&\cdots&0\\0&0&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\0&0&\cdots&1\\ -P_n(t)/P_0(t)&-P_{n-1}/P_0(t)& \cdots&-P_1(t)/P_0(t)\end{array} \right]{\bf y}+ \left[\begin{array}{c} 0\\0\\\vdots\\F(t)/P_0(t)\end{array}\right].}\)

(b) \({\dst{\left[\begin{array}{cccc} y_1&y_2&\cdots&y_n\\y_1'&y_2'&\cdots&y_n'\\ \vdots&\vdots&\ddots&\vdots\\y_1^{(n-1)}&y_2^{(n-1)}&\cdots&y_n^{(n-1)} \end{array}\right]}}\)