Internal problem ID [6742]
Internal file name [OUTPUT/5990_Sunday_June_05_2022_04_11_24_PM_42613464/index.tex]
Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL,
WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th
edition.
Section: CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS.
EXERCISES 8.2. Page 346
Problem number: 15.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "system of linear ODEs"
Solve \begin {align*} x^{\prime }\left (t \right )&=\frac {9 x \left (t \right )}{10}+\frac {21 y}{10}+\frac {16 z \left (t \right )}{5}\\ y^{\prime }&=\frac {7 x \left (t \right )}{10}+\frac {13 y}{2}+\frac {21 z \left (t \right )}{5}\\ z^{\prime }\left (t \right )&=\frac {11 x \left (t \right )}{10}+\frac {17 y}{10}+\frac {17 z \left (t \right )}{5} \end {align*}
In this method, we will assume we have found the matrix exponential \(e^{A t}\) allready. There are different methods to determine this but will not be shown here. This is a system of linear ODE’s given as Warning. Unable to find the matrix exponential.
This is a system of linear ODE’s given as \begin {align*} \vec {x}'(t) &= A\, \vec {x}(t) \end {align*}
Or \begin {align*} \left [\begin {array}{c} x^{\prime }\left (t \right ) \\ y^{\prime } \\ z^{\prime }\left (t \right ) \end {array}\right ] &= \left [\begin {array}{ccc} \frac {9}{10} & \frac {21}{10} & \frac {16}{5} \\ \frac {7}{10} & \frac {13}{2} & \frac {21}{5} \\ \frac {11}{10} & \frac {17}{10} & \frac {17}{5} \end {array}\right ]\, \left [\begin {array}{c} x \left (t \right ) \\ y \\ z \left (t \right ) \end {array}\right ] \end {align*}
The first step is find the homogeneous solution. We start by finding the eigenvalues of \(A\). This is done by solving the following equation for the eigenvalues \(\lambda \) \begin {align*} \operatorname {det} \left ( A- \lambda I \right ) &= 0 \end {align*}
Expanding gives \begin {align*} \operatorname {det} \left (\left [\begin {array}{ccc} \frac {9}{10} & \frac {21}{10} & \frac {16}{5} \\ \frac {7}{10} & \frac {13}{2} & \frac {21}{5} \\ \frac {11}{10} & \frac {17}{10} & \frac {17}{5} \end {array}\right ]-\lambda \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) &= 0 \end {align*}
Therefore \begin {align*} \operatorname {det} \left (\left [\begin {array}{ccc} \frac {9}{10}-\lambda & \frac {21}{10} & \frac {16}{5} \\ \frac {7}{10} & \frac {13}{2}-\lambda & \frac {21}{5} \\ \frac {11}{10} & \frac {17}{10} & \frac {17}{5}-\lambda \end {array}\right ]\right ) &= 0 \end {align*}
Which gives the characteristic equation \begin {align*} \lambda ^{3}-\frac {54}{5} \lambda ^{2}+\frac {472}{25} \lambda +\frac {113}{125}&=0 \end {align*}
The roots of the above are the eigenvalues. \begin {align*} \lambda _1 &= \frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\\ \lambda _2 &= -\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\\ \lambda _3 &= -\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10} \end {align*}
This table summarises the above result
| eigenvalue | algebraic multiplicity | type of eigenvalue |
| \(\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\) | \(1\) | complex eigenvalue |
| \(-\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\) | \(1\) | complex eigenvalue |
| \(-\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\) | \(1\) | complex eigenvalue |
Now the eigenvector for each eigenvalue are found.
Considering the eigenvalue \(\lambda _{1} = \frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\)
We need to solve \(A \vec {v} = \lambda \vec {v}\) or \((A-\lambda I) \vec {v} = \vec {0}\) which becomes \begin {align*} \left (\left [\begin {array}{ccc} \frac {9}{10} & \frac {21}{10} & \frac {16}{5} \\ \frac {7}{10} & \frac {13}{2} & \frac {21}{5} \\ \frac {11}{10} & \frac {17}{10} & \frac {17}{5} \end {array}\right ] - \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right ) \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccc} \frac {-\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-81 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6000}{30 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}} & \frac {21}{10} & \frac {16}{5} \\ \frac {7}{10} & \frac {-\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+87 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6000}{30 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}} & \frac {21}{5} \\ \frac {11}{10} & \frac {17}{10} & \frac {-\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6000}{30 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}} \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \end {align*}
Now forward elimination is applied to solve for the eigenvector \(\vec {v}\). The augmented matrix is \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {27}{10}-\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{30}-\frac {200}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}&{\frac {21}{10}}&{\frac {16}{5}}&0\\ {\frac {7}{10}}&\frac {29}{10}-\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{30}-\frac {200}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}&{\frac {21}{5}}&0\\ {\frac {11}{10}}&{\frac {17}{10}}&-\frac {1}{5}-\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{30}-\frac {200}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}-\frac {7 R_{1}}{10 \left (-\frac {27}{10}-\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{30}-\frac {200}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {-\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-81 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6000}{30 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}&{\frac {21}{10}}&{\frac {16}{5}}&0\\ 0&-\frac {2 \left (\left (i \sqrt {29760999}+4899\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6 i \sqrt {29760999}+\frac {121 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}{2}+567006\right )}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \left (\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+81 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right )}&\frac {21 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+2037 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+126000}{5 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+405 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+30000}&0\\ {\frac {11}{10}}&{\frac {17}{10}}&\frac {-\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6000}{30 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}-\frac {33 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} R_{1}}{-\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-81 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6000} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {-\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-81 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6000}{30 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}&{\frac {21}{10}}&{\frac {16}{5}}&0\\ 0&-\frac {2 \left (\left (i \sqrt {29760999}+4899\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6 i \sqrt {29760999}+\frac {121 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}{2}+567006\right )}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \left (\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+81 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right )}&\frac {21 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+2037 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+126000}{5 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+405 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+30000}&0\\ 0&\frac {17 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+2070 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+102000}{10 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+810 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+60000}&-\frac {2 \left (\left (i \sqrt {29760999}+14199\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+87 i \sqrt {29760999}+\frac {1553 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}{10}+1078413\right )}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \left (\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+81 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right )}&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}+\frac {\left (17 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+2070 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+102000\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \left (\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+81 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right ) R_{2}}{2 \left (10 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+810 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+60000\right ) \left (\left (i \sqrt {29760999}+4899\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6 i \sqrt {29760999}+\frac {121 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}{2}+567006\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {-\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-81 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6000}{30 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}&{\frac {21}{10}}&{\frac {16}{5}}&0\\ 0&-\frac {2 \left (\left (i \sqrt {29760999}+4899\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6 i \sqrt {29760999}+\frac {121 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}{2}+567006\right )}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \left (\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+81 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right )}&\frac {21 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+2037 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+126000}{5 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+405 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+30000}&0\\ 0&0&0&0 \end {array} \right ] \end {align*}
Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} \frac {-\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-81 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6000}{30 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}} & \frac {21}{10} & \frac {16}{5} \\ 0 & -\frac {2 \left (\left (i \sqrt {29760999}+4899\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6 i \sqrt {29760999}+\frac {121 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}{2}+567006\right )}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \left (\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+81 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right )} & \frac {21 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+2037 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+126000}{5 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+405 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+30000} \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are \(\{v_{3}\}\) and the leading variables are \(\{v_{1}, v_{2}\}\). Let \(v_{3} = t\). Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation \(\left \{v_{1} = \frac {3 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} t \left (320 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \sqrt {29760999}+24540 i \sqrt {29760999}+62137 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+4213680 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+326945460\right )}{5 \left (2 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \sqrt {29760999}-12 i \sqrt {29760999}+121 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+9798 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+1134012\right ) \left (\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+81 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right )}, v_{2} = \frac {21 t \left (60 i \sqrt {29760999}+97 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+6000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+329940\right )}{5 \left (2 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \sqrt {29760999}-12 i \sqrt {29760999}+121 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+9798 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+1134012\right )}\right \}\)
Hence the solution is \[ \left [\begin {array}{c} \frac {3 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}} t \left (320 \,\operatorname {I} \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}} \sqrt {29760999}+24540 \,\operatorname {I} \sqrt {29760999}+62137 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {2}{3}}+4213680 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}+326945460\right )}{5 \left (2 \,\operatorname {I} \sqrt {29760999}\, \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}-12 \,\operatorname {I} \sqrt {29760999}+121 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {2}{3}}+9798 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}+1134012\right ) \left (\left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {2}{3}}+81 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right )} \\ \frac {21 t \left (60 \,\operatorname {I} \sqrt {29760999}+97 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {2}{3}}+6000 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}+329940\right )}{5 \left (2 \,\operatorname {I} \sqrt {29760999}\, \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}-12 \,\operatorname {I} \sqrt {29760999}+121 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {2}{3}}+9798 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}+1134012\right )} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {3 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} t \left (320 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \sqrt {29760999}+24540 i \sqrt {29760999}+62137 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+4213680 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+326945460\right )}{5 \left (2 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \sqrt {29760999}-12 i \sqrt {29760999}+121 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+9798 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+1134012\right ) \left (\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+81 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right )} \\ \frac {21 t \left (60 i \sqrt {29760999}+97 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+6000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+329940\right )}{5 \left (2 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \sqrt {29760999}-12 i \sqrt {29760999}+121 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+9798 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+1134012\right )} \\ t \end {array}\right ] \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \left [\begin {array}{c} \frac {3 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}} t \left (320 \,\operatorname {I} \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}} \sqrt {29760999}+24540 \,\operatorname {I} \sqrt {29760999}+62137 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {2}{3}}+4213680 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}+326945460\right )}{5 \left (2 \,\operatorname {I} \sqrt {29760999}\, \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}-12 \,\operatorname {I} \sqrt {29760999}+121 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {2}{3}}+9798 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}+1134012\right ) \left (\left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {2}{3}}+81 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right )} \\ \frac {21 t \left (60 \,\operatorname {I} \sqrt {29760999}+97 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {2}{3}}+6000 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}+329940\right )}{5 \left (2 \,\operatorname {I} \sqrt {29760999}\, \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}-12 \,\operatorname {I} \sqrt {29760999}+121 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {2}{3}}+9798 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}+1134012\right )} \\ t \end {array}\right ] = t \left [\begin {array}{c} \frac {3 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \left (320 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \sqrt {29760999}+24540 i \sqrt {29760999}+62137 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+4213680 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+326945460\right )}{5 \left (2 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \sqrt {29760999}-12 i \sqrt {29760999}+121 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+9798 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+1134012\right ) \left (\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+81 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right )} \\ \frac {252 i \sqrt {29760999}+\frac {2037 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}{5}+25200 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+1385748}{2 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \sqrt {29760999}-12 i \sqrt {29760999}+121 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+9798 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+1134012} \\ 1 \end {array}\right ] \] Let \(t = 1\) the eigenvector becomes \[ \left [\begin {array}{c} \frac {3 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}} t \left (320 \,\operatorname {I} \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}} \sqrt {29760999}+24540 \,\operatorname {I} \sqrt {29760999}+62137 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {2}{3}}+4213680 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}+326945460\right )}{5 \left (2 \,\operatorname {I} \sqrt {29760999}\, \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}-12 \,\operatorname {I} \sqrt {29760999}+121 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {2}{3}}+9798 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}+1134012\right ) \left (\left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {2}{3}}+81 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right )} \\ \frac {21 t \left (60 \,\operatorname {I} \sqrt {29760999}+97 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {2}{3}}+6000 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}+329940\right )}{5 \left (2 \,\operatorname {I} \sqrt {29760999}\, \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}-12 \,\operatorname {I} \sqrt {29760999}+121 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {2}{3}}+9798 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}+1134012\right )} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {3 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \left (320 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \sqrt {29760999}+24540 i \sqrt {29760999}+62137 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+4213680 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+326945460\right )}{5 \left (2 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \sqrt {29760999}-12 i \sqrt {29760999}+121 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+9798 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+1134012\right ) \left (\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+81 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right )} \\ \frac {252 i \sqrt {29760999}+\frac {2037 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}{5}+25200 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+1385748}{2 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \sqrt {29760999}-12 i \sqrt {29760999}+121 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+9798 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+1134012} \\ 1 \end {array}\right ] \] Which is normalized to \[ \left [\begin {array}{c} \frac {3 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}} t \left (320 \,\operatorname {I} \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}} \sqrt {29760999}+24540 \,\operatorname {I} \sqrt {29760999}+62137 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {2}{3}}+4213680 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}+326945460\right )}{5 \left (2 \,\operatorname {I} \sqrt {29760999}\, \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}-12 \,\operatorname {I} \sqrt {29760999}+121 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {2}{3}}+9798 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}+1134012\right ) \left (\left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {2}{3}}+81 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right )} \\ \frac {21 t \left (60 \,\operatorname {I} \sqrt {29760999}+97 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {2}{3}}+6000 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}+329940\right )}{5 \left (2 \,\operatorname {I} \sqrt {29760999}\, \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}-12 \,\operatorname {I} \sqrt {29760999}+121 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {2}{3}}+9798 \left (329940+60 \,\operatorname {I} \sqrt {29760999}\right )^{\frac {1}{3}}+1134012\right )} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {3 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \left (320 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \sqrt {29760999}+24540 i \sqrt {29760999}+62137 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+4213680 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+326945460\right )}{5 \left (2 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \sqrt {29760999}-12 i \sqrt {29760999}+121 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+9798 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+1134012\right ) \left (\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+81 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right )} \\ \frac {252 i \sqrt {29760999}+\frac {2037 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}{5}+25200 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+1385748}{2 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} \sqrt {29760999}-12 i \sqrt {29760999}+121 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+9798 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+1134012} \\ 1 \end {array}\right ] \] Considering the eigenvalue \(\lambda _{2} = -\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\)
We need to solve \(A \vec {v} = \lambda \vec {v}\) or \((A-\lambda I) \vec {v} = \vec {0}\) which becomes \begin {align*} \left (\left [\begin {array}{ccc} \frac {9}{10} & \frac {21}{10} & \frac {16}{5} \\ \frac {7}{10} & \frac {13}{2} & \frac {21}{5} \\ \frac {11}{10} & \frac {17}{10} & \frac {17}{5} \end {array}\right ] - \left (-\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right ) \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccc} -\frac {27}{10}+\frac {2 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )}{3}-2 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5} & \frac {21}{10} & \frac {16}{5} \\ \frac {7}{10} & \frac {29}{10}+\frac {2 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )}{3}-2 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5} & \frac {21}{5} \\ \frac {11}{10} & \frac {17}{10} & -\frac {1}{5}+\frac {2 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )}{3}-2 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5} \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \end {align*}
Now forward elimination is applied to solve for the eigenvector \(\vec {v}\). The augmented matrix is \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {27}{10}+\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}+\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}&{\frac {21}{10}}&{\frac {16}{5}}&0\\ {\frac {7}{10}}&\frac {29}{10}+\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}+\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}&{\frac {21}{5}}&0\\ {\frac {11}{10}}&{\frac {17}{10}}&-\frac {1}{5}+\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}+\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}-\frac {7 R_{1}}{10 \left (-\frac {27}{10}+\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}+\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {27}{10}+\frac {2 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )}{3}-2 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}&{\frac {21}{10}}&{\frac {16}{5}}&0\\ 0&\frac {2 \left (4899 i \sqrt {3}-i \sqrt {29760999}-3 \sqrt {9920333}-4899\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-1134012 i \sqrt {3}+12 i \sqrt {29760999}-36 \sqrt {9920333}+242 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-1134012}{\left (-162 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+\left (1+i \sqrt {3}\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}+6000\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}&\frac {-25200 i+\frac {21 \left (329940+60 i \sqrt {3}\, \sqrt {9920333}\right )^{\frac {2}{3}} \left (-i+\sqrt {3}\right )}{5}-25200 \sqrt {3}+\frac {4074 i \left (329940+60 i \sqrt {3}\, \sqrt {9920333}\right )^{\frac {1}{3}}}{5}}{-6000 i+\left (329940+60 i \sqrt {3}\, \sqrt {9920333}\right )^{\frac {2}{3}} \left (-i+\sqrt {3}\right )-6000 \sqrt {3}+162 i \left (329940+60 i \sqrt {3}\, \sqrt {9920333}\right )^{\frac {1}{3}}}&0\\ {\frac {11}{10}}&{\frac {17}{10}}&-\frac {1}{5}+\frac {2 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )}{3}-2 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}-\frac {11 R_{1}}{10 \left (-\frac {27}{10}+\frac {2 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )}{3}-2 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {27}{10}+\frac {2 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )}{3}-2 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}&{\frac {21}{10}}&{\frac {16}{5}}&0\\ 0&\frac {2 \left (4899 i \sqrt {3}-i \sqrt {29760999}-3 \sqrt {9920333}-4899\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-1134012 i \sqrt {3}+12 i \sqrt {29760999}-36 \sqrt {9920333}+242 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-1134012}{\left (-162 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+\left (1+i \sqrt {3}\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}+6000\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}&\frac {-25200 i+\frac {21 \left (329940+60 i \sqrt {3}\, \sqrt {9920333}\right )^{\frac {2}{3}} \left (-i+\sqrt {3}\right )}{5}-25200 \sqrt {3}+\frac {4074 i \left (329940+60 i \sqrt {3}\, \sqrt {9920333}\right )^{\frac {1}{3}}}{5}}{-6000 i+\left (329940+60 i \sqrt {3}\, \sqrt {9920333}\right )^{\frac {2}{3}} \left (-i+\sqrt {3}\right )-6000 \sqrt {3}+162 i \left (329940+60 i \sqrt {3}\, \sqrt {9920333}\right )^{\frac {1}{3}}}&0\\ 0&\frac {34 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )-102 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}-207}{20 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )-60 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}-81}&\frac {1553-1000 \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{3}\right )-1000 \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{3}\right )-290 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )+870 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}}{100 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )-300 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}-405}&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}-\frac {\left (34 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )-102 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}-207\right ) \left (-162 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+\left (1+i \sqrt {3}\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}+6000\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} R_{2}}{2 \left (20 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )-60 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}-81\right ) \left (\left (4899 i \sqrt {3}-i \sqrt {29760999}-3 \sqrt {9920333}-4899\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-567006 i \sqrt {3}+6 i \sqrt {29760999}-18 \sqrt {9920333}+121 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-567006\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {27}{10}+\frac {2 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )}{3}-2 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}&{\frac {21}{10}}&{\frac {16}{5}}&0\\ 0&\frac {2 \left (4899 i \sqrt {3}-i \sqrt {29760999}-3 \sqrt {9920333}-4899\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-1134012 i \sqrt {3}+12 i \sqrt {29760999}-36 \sqrt {9920333}+242 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-1134012}{\left (-162 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+\left (1+i \sqrt {3}\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}+6000\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}&\frac {-25200 i+\frac {21 \left (329940+60 i \sqrt {3}\, \sqrt {9920333}\right )^{\frac {2}{3}} \left (-i+\sqrt {3}\right )}{5}-25200 \sqrt {3}+\frac {4074 i \left (329940+60 i \sqrt {3}\, \sqrt {9920333}\right )^{\frac {1}{3}}}{5}}{-6000 i+\left (329940+60 i \sqrt {3}\, \sqrt {9920333}\right )^{\frac {2}{3}} \left (-i+\sqrt {3}\right )-6000 \sqrt {3}+162 i \left (329940+60 i \sqrt {3}\, \sqrt {9920333}\right )^{\frac {1}{3}}}&0\\ 0&0&\text {Expression too large to display}&0 \end {array} \right ] \end {align*}
Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} -\frac {27}{10}+\frac {2 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )}{3}-2 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5} & \frac {21}{10} & \frac {16}{5} \\ 0 & \frac {2 \left (4899 i \sqrt {3}-i \sqrt {29760999}-3 \sqrt {9920333}-4899\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-1134012 i \sqrt {3}+12 i \sqrt {29760999}-36 \sqrt {9920333}+242 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-1134012}{\left (-162 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+\left (1+i \sqrt {3}\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}+6000\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}} & \frac {-25200 i+\frac {21 \left (329940+60 i \sqrt {3}\, \sqrt {9920333}\right )^{\frac {2}{3}} \left (-i+\sqrt {3}\right )}{5}-25200 \sqrt {3}+\frac {4074 i \left (329940+60 i \sqrt {3}\, \sqrt {9920333}\right )^{\frac {1}{3}}}{5}}{-6000 i+\left (329940+60 i \sqrt {3}\, \sqrt {9920333}\right )^{\frac {2}{3}} \left (-i+\sqrt {3}\right )-6000 \sqrt {3}+162 i \left (329940+60 i \sqrt {3}\, \sqrt {9920333}\right )^{\frac {1}{3}}} \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are \(\{v_{3}\}\) and the leading variables are \(\{v_{1}, v_{2}\}\). Let \(v_{3} = t\). Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation \(\text {Expression too large to display}\)
Hence the solution is \[ \text {Expression too large to display} \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \text {Expression too large to display} \text {Expression too large to display} \] Let \(t = 1\) the eigenvector becomes \[ \text {Expression too large to display} \] Which is normalized to \[ \text {Expression too large to display} \] Considering the eigenvalue \(\lambda _{3} = -\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\)
We need to solve \(A \vec {v} = \lambda \vec {v}\) or \((A-\lambda I) \vec {v} = \vec {0}\) which becomes \begin {align*} \left (\left [\begin {array}{ccc} \frac {9}{10} & \frac {21}{10} & \frac {16}{5} \\ \frac {7}{10} & \frac {13}{2} & \frac {21}{5} \\ \frac {11}{10} & \frac {17}{10} & \frac {17}{5} \end {array}\right ] - \left (-\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right ) \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccc} -\frac {27}{10}+\frac {2 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )}{3}+2 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5} & \frac {21}{10} & \frac {16}{5} \\ \frac {7}{10} & \frac {29}{10}+\frac {2 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )}{3}+2 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5} & \frac {21}{5} \\ \frac {11}{10} & \frac {17}{10} & -\frac {1}{5}+\frac {2 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )}{3}+2 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5} \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \end {align*}
Now forward elimination is applied to solve for the eigenvector \(\vec {v}\). The augmented matrix is \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {27}{10}+\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}+\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}&{\frac {21}{10}}&{\frac {16}{5}}&0\\ {\frac {7}{10}}&\frac {29}{10}+\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}+\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}&{\frac {21}{5}}&0\\ {\frac {11}{10}}&{\frac {17}{10}}&-\frac {1}{5}+\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}+\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}-\frac {7 R_{1}}{10 \left (-\frac {27}{10}+\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}+\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {27}{10}+\frac {2 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )}{3}+2 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}&{\frac {21}{10}}&{\frac {16}{5}}&0\\ 0&\frac {2 \left (4899 i \sqrt {3}+i \sqrt {29760999}-3 \sqrt {9920333}+4899\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-1134012 i \sqrt {3}-12 i \sqrt {29760999}-36 \sqrt {9920333}-242 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+1134012}{\left (162 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+\left (i \sqrt {3}-1\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}-6000\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}&\frac {-25200 i-\frac {21 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )}{5}+25200 \sqrt {3}+\frac {4074 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{5}}{-6000 i-\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )+6000 \sqrt {3}+162 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}&0\\ {\frac {11}{10}}&{\frac {17}{10}}&-\frac {1}{5}+\frac {2 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )}{3}+2 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}-\frac {11 R_{1}}{10 \left (-\frac {27}{10}+\frac {2 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )}{3}+2 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {27}{10}+\frac {2 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )}{3}+2 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}&{\frac {21}{10}}&{\frac {16}{5}}&0\\ 0&\frac {2 \left (4899 i \sqrt {3}+i \sqrt {29760999}-3 \sqrt {9920333}+4899\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-1134012 i \sqrt {3}-12 i \sqrt {29760999}-36 \sqrt {9920333}-242 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+1134012}{\left (162 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+\left (i \sqrt {3}-1\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}-6000\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}&\frac {-25200 i-\frac {21 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )}{5}+25200 \sqrt {3}+\frac {4074 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{5}}{-6000 i-\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )+6000 \sqrt {3}+162 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}&0\\ 0&\frac {34 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )+102 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}-207}{20 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )+60 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}-81}&\frac {1553-1000 \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{3}\right )+1000 \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{3}\right )-290 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )-870 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}}{100 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )+300 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}-405}&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}-\frac {\left (34 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )+102 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}-207\right ) \left (162 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+\left (i \sqrt {3}-1\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}-6000\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}} R_{2}}{2 \left (20 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )+60 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}-81\right ) \left (\left (4899 i \sqrt {3}+i \sqrt {29760999}-3 \sqrt {9920333}+4899\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-567006 i \sqrt {3}-6 i \sqrt {29760999}-18 \sqrt {9920333}-121 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+567006\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {27}{10}+\frac {2 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )}{3}+2 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}&{\frac {21}{10}}&{\frac {16}{5}}&0\\ 0&\frac {2 \left (4899 i \sqrt {3}+i \sqrt {29760999}-3 \sqrt {9920333}+4899\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-1134012 i \sqrt {3}-12 i \sqrt {29760999}-36 \sqrt {9920333}-242 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+1134012}{\left (162 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+\left (i \sqrt {3}-1\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}-6000\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}&\frac {-25200 i-\frac {21 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )}{5}+25200 \sqrt {3}+\frac {4074 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{5}}{-6000 i-\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )+6000 \sqrt {3}+162 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}&0\\ 0&0&\frac {730441728+\left (329940+60 i \sqrt {3}\, \sqrt {9920333}\right )^{\frac {1}{3}} \left (-533853-9798000 \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )^{2}+870 \left (-\left (-\sqrt {9920333}-\frac {24043}{29}\right ) \sqrt {3}-i \sqrt {9920333}+\frac {72129 i}{29}\right ) \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )+870 \sqrt {5}\, \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \left (\frac {72129}{29}-i \left (\sqrt {9920333}-\frac {72129}{29}\right ) \sqrt {3}+3 \sqrt {9920333}\right )+3000 \left (4899 i+\left (-\sqrt {9920333}+1633\right ) \sqrt {3}+i \sqrt {9920333}\right ) \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{3}\right )+1000 \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{3}\right ) \left (i \left (-4899-\sqrt {9920333}\right ) \sqrt {3}+3 \sqrt {9920333}\right )+i \left (-5432853+1553 \sqrt {9920333}\right ) \sqrt {3}-4659 \sqrt {9920333}\right )+2 \left (-17084 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )-60500 \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{3}\right )-51252 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}+60500 \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{3}\right )+116873\right ) \left (329940+60 i \sqrt {3}\, \sqrt {9920333}\right )^{\frac {2}{3}}-1134012000 \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )^{2}+69480 \sqrt {5}\, \left (\left (\sqrt {9920333}-\frac {388693}{579}\right ) \sqrt {3}+i \sqrt {9920333}+\frac {388693 i}{193}\right ) \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )+69480 \sqrt {5}\, \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \left (-\frac {388693}{193}+i \left (\sqrt {9920333}+\frac {388693}{193}\right ) \sqrt {3}+3 \sqrt {9920333}\right )+6000 \left (-283503 i+\left (-3 \sqrt {9920333}+94501\right ) \sqrt {3}-3 i \sqrt {9920333}\right ) \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{3}\right )+6000 \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{3}\right ) \left (i \left (\sqrt {9920333}+94501\right ) \sqrt {3}+3 \sqrt {9920333}\right )+48 i \left (-2911 \sqrt {9920333}-3404911\right ) \sqrt {3}-419184 \sqrt {9920333}}{100 \left (\sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )+3 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5}-\frac {81}{20}\right ) \left (567006+\left (329940+60 i \sqrt {3}\, \sqrt {9920333}\right )^{\frac {1}{3}} \left (4899+i \left (\sqrt {9920333}+4899\right ) \sqrt {3}-3 \sqrt {9920333}\right )-121 \left (329940+60 i \sqrt {3}\, \sqrt {9920333}\right )^{\frac {2}{3}}+6 i \left (-\sqrt {9920333}-94501\right ) \sqrt {3}-18 \sqrt {9920333}\right )}&0 \end {array} \right ] \end {align*}
Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} -\frac {27}{10}+\frac {2 \sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right )}{3}+2 \sin \left (\frac {\arctan \left (\frac {1833 \sqrt {29760999}}{79667}\right )}{6}\right ) \sqrt {5} & \frac {21}{10} & \frac {16}{5} \\ 0 & \frac {2 \left (4899 i \sqrt {3}+i \sqrt {29760999}-3 \sqrt {9920333}+4899\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-1134012 i \sqrt {3}-12 i \sqrt {29760999}-36 \sqrt {9920333}-242 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+1134012}{\left (162 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+\left (i \sqrt {3}-1\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}-6000\right ) \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}} & \frac {-25200 i-\frac {21 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )}{5}+25200 \sqrt {3}+\frac {4074 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{5}}{-6000 i-\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )+6000 \sqrt {3}+162 i \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}} \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are \(\{v_{3}\}\) and the leading variables are \(\{v_{1}, v_{2}\}\). Let \(v_{3} = t\). Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation \(\text {Expression too large to display}\)
Hence the solution is \[ \text {Expression too large to display} \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \text {Expression too large to display} \text {Expression too large to display} \] Let \(t = 1\) the eigenvector becomes \[ \text {Expression too large to display} \] Which is normalized to \[ \text {Expression too large to display} \] The following table gives a summary of this result. It shows for each eigenvalue the algebraic multiplicity \(m\), and its geometric multiplicity \(k\) and the eigenvectors associated with the eigenvalue. If \(m>k\) then the eigenvalue is defective which means the number of normal linearly independent eigenvectors associated with this eigenvalue (called the geometric multiplicity \(k\)) does not equal the algebraic multiplicity \(m\), and we need to determine an additional \(m-k\) generalized eigenvectors for this eigenvalue.
| multiplicity
| ||||
| eigenvalue | algebraic \(m\) | geometric \(k\) | defective? | eigenvectors |
| \(\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\) | \(1\) | \(1\) | No | \(\left [\begin {array}{c} \frac {\frac {22576 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {45152000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}-32453}{\left (-\frac {334 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{3}-\frac {668000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}-47413+1050 \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right )^{2}\right ) \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {2000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+27\right )} \\ -\frac {1600 \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right )^{2}-528 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-\frac {3168000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}-70703}{-\frac {334 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{3}-\frac {668000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}-47413+1050 \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right )^{2}} \\ 1 \end {array}\right ]\) |
| \(-\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\) | \(1\) | \(1\) | No | \(\left [\begin {array}{c} \frac {-\frac {11288 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{3}-\frac {22576000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}-32453+22576 i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{\left (\frac {167 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {334000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}-47413-334 i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )+1050 \left (-\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}\right ) \left (-\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+27+i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )\right )} \\ -\frac {1600 \left (-\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}+264 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+\frac {1584000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}-70703-1584 i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{\frac {167 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {334000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}-47413-334 i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )+1050 \left (-\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}} \\ 1 \end {array}\right ]\) |
| \(-\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\) | \(1\) | \(1\) | No | \(\left [\begin {array}{c} \frac {-\frac {11288 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{3}-\frac {22576000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}-32453-22576 i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{\left (\frac {167 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {334000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}-47413+334 i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )+1050 \left (-\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}\right ) \left (-\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+27-i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )\right )} \\ -\frac {1600 \left (-\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}+264 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+\frac {1584000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}-70703+1584 i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{\frac {167 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {334000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}-47413+334 i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )+1050 \left (-\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {i \sqrt {3}\, \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}} \\ 1 \end {array}\right ]\) |
Now that we found the eigenvalues and associated eigenvectors, we will go over each eigenvalue and generate the solution basis. The only problem we need to take care of is if the eigenvalue is defective. Therefore the final solution is \begin {align*} \vec {x}_h(t) &= c_{1} \vec {x}_{1}(t) + c_{2} \vec {x}_{2}(t) + c_{3} \vec {x}_{3}(t) \end {align*}
Which is written as \begin {align*} \left [\begin {array}{c} x \left (t \right ) \\ y \\ z \left (t \right ) \end {array}\right ] &= c_{1} \left [\begin {array}{c} \frac {1411 \,{\mathrm e}^{\left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right ) t} \left (\frac {16 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {32000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}-23\right )}{\left (-\frac {334 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{3}-\frac {668000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}-47413+1050 \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right )^{2}\right ) \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {2000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+27\right )} \\ -\frac {{\mathrm e}^{\left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right ) t} \left (1600 \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right )^{2}-528 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-\frac {3168000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}-70703\right )}{-\frac {334 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{3}-\frac {668000}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}-47413+1050 \left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right )^{2}} \\ {\mathrm e}^{\left (\frac {\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right ) t} \end {array}\right ] + c_{2} \text {Expression too large to display} + c_{3} \text {Expression too large to display} \end {align*}
Which becomes \begin {align*} \text {Expression too large to display} \end {align*}
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [x^{\prime }\left (t \right )=\frac {9 x \left (t \right )}{10}+\frac {21 y}{10}+\frac {16 z \left (t \right )}{5}, y^{\prime }=\frac {7 x \left (t \right )}{10}+\frac {13 y}{2}+\frac {21 z \left (t \right )}{5}, z^{\prime }\left (t \right )=\frac {11 x \left (t \right )}{10}+\frac {17 y}{10}+\frac {17 z \left (t \right )}{5}\right ] \\ \bullet & {} & \textrm {Define vector}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}\left (t \right )=\left [\begin {array}{c} x \left (t \right ) \\ y \\ z \left (t \right ) \end {array}\right ] \\ \bullet & {} & \textrm {Convert system into a vector equation}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}^{\prime }\left (t \right )=\left [\begin {array}{ccc} \frac {9}{10} & \frac {21}{10} & \frac {16}{5} \\ \frac {7}{10} & \frac {13}{2} & \frac {21}{5} \\ \frac {11}{10} & \frac {17}{10} & \frac {17}{5} \end {array}\right ]\cdot {\moverset {\rightarrow }{x}}\left (t \right )+\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}^{\prime }\left (t \right )=\left [\begin {array}{ccc} \frac {9}{10} & \frac {21}{10} & \frac {16}{5} \\ \frac {7}{10} & \frac {13}{2} & \frac {21}{5} \\ \frac {11}{10} & \frac {17}{10} & \frac {17}{5} \end {array}\right ]\cdot {\moverset {\rightarrow }{x}}\left (t \right ) \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{ccc} \frac {9}{10} & \frac {21}{10} & \frac {16}{5} \\ \frac {7}{10} & \frac {13}{2} & \frac {21}{5} \\ \frac {11}{10} & \frac {17}{10} & \frac {17}{5} \end {array}\right ] \\ \bullet & {} & \textrm {Rewrite the system as}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}^{\prime }\left (t \right )=A \cdot {\moverset {\rightarrow }{x}}\left (t \right ) \\ \bullet & {} & \textrm {To solve the system, find the eigenvalues and eigenvectors of}\hspace {3pt} A \\ \bullet & {} & \textrm {Eigenpairs of}\hspace {3pt} A \\ {} & {} & \left [\left [\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}, \left [\begin {array}{c} \frac {1411 \left (\frac {16 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {32000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-23\right )}{\left (-\frac {334 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}-\frac {668000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413+1050 \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right )^{2}\right ) \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {2000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+27\right )} \\ -\frac {1600 \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right )^{2}-528 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}-\frac {3168000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-70703}{-\frac {334 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}-\frac {668000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413+1050 \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right )^{2}} \\ 1 \end {array}\right ]\right ], \left [-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}, \left [\begin {array}{c} \frac {1411 \left (-\frac {8 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}-\frac {16000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-23-16 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )\right )}{\left (\frac {167 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {334000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413+334 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )+1050 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}\right ) \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+27-\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )\right )} \\ -\frac {1600 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}+264 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}+\frac {1584000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-70703+1584 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{\frac {167 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {334000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413+334 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )+1050 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}} \\ 1 \end {array}\right ]\right ], \left [-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}, \left [\begin {array}{c} \frac {1411 \left (-\frac {8 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}-\frac {16000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-23+16 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )\right )}{\left (\frac {167 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {334000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413-334 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )+1050 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}\right ) \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+27+\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )\right )} \\ -\frac {1600 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}+264 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}+\frac {1584000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-70703-1584 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{\frac {167 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {334000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413-334 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )+1050 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}, \left [\begin {array}{c} \frac {1411 \left (\frac {16 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {32000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-23\right )}{\left (-\frac {334 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}-\frac {668000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413+1050 \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right )^{2}\right ) \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {2000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+27\right )} \\ -\frac {1600 \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right )^{2}-528 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}-\frac {3168000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-70703}{-\frac {334 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}-\frac {668000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413+1050 \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right )^{2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}_{1}={\mathrm e}^{\left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right ) t}\cdot \left [\begin {array}{c} \frac {1411 \left (\frac {16 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {32000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-23\right )}{\left (-\frac {334 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}-\frac {668000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413+1050 \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right )^{2}\right ) \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {2000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+27\right )} \\ -\frac {1600 \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right )^{2}-528 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}-\frac {3168000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-70703}{-\frac {334 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}-\frac {668000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413+1050 \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right )^{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}, \left [\begin {array}{c} \frac {1411 \left (-\frac {8 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}-\frac {16000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-23-16 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )\right )}{\left (\frac {167 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {334000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413+334 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )+1050 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}\right ) \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+27-\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )\right )} \\ -\frac {1600 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}+264 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}+\frac {1584000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-70703+1584 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{\frac {167 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {334000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413+334 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )+1050 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}_{2}={\mathrm e}^{\left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right ) t}\cdot \left [\begin {array}{c} \frac {1411 \left (-\frac {8 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}-\frac {16000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-23-16 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )\right )}{\left (\frac {167 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {334000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413+334 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )+1050 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}\right ) \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+27-\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )\right )} \\ -\frac {1600 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}+264 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}+\frac {1584000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-70703+1584 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{\frac {167 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {334000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413+334 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )+1050 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}, \left [\begin {array}{c} \frac {1411 \left (-\frac {8 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}-\frac {16000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-23+16 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )\right )}{\left (\frac {167 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {334000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413-334 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )+1050 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}\right ) \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+27+\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )\right )} \\ -\frac {1600 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}+264 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}+\frac {1584000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-70703-1584 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{\frac {167 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {334000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413-334 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )+1050 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}_{3}={\mathrm e}^{\left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right ) t}\cdot \left [\begin {array}{c} \frac {1411 \left (-\frac {8 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}-\frac {16000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-23+16 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )\right )}{\left (\frac {167 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {334000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413-334 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )+1050 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}\right ) \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+27+\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )\right )} \\ -\frac {1600 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}+264 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}+\frac {1584000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-70703-1584 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{\frac {167 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {334000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413-334 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )+1050 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {General solution to the system of ODEs}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}=c_{1} {\moverset {\rightarrow }{x}}_{1}+c_{2} {\moverset {\rightarrow }{x}}_{2}+c_{3} {\moverset {\rightarrow }{x}}_{3} \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}=c_{1} {\mathrm e}^{\left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right ) t}\cdot \left [\begin {array}{c} \frac {1411 \left (\frac {16 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {32000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-23\right )}{\left (-\frac {334 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}-\frac {668000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413+1050 \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right )^{2}\right ) \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {2000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+27\right )} \\ -\frac {1600 \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right )^{2}-528 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}-\frac {3168000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-70703}{-\frac {334 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}-\frac {668000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413+1050 \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{30}+\frac {200}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}\right )^{2}} \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{\left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right ) t}\cdot \left [\begin {array}{c} \frac {1411 \left (-\frac {8 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}-\frac {16000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-23-16 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )\right )}{\left (\frac {167 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {334000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413+334 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )+1050 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}\right ) \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+27-\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )\right )} \\ -\frac {1600 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}+264 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}+\frac {1584000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-70703+1584 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{\frac {167 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {334000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413+334 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )+1050 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}} \\ 1 \end {array}\right ]+c_{3} {\mathrm e}^{\left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right ) t}\cdot \left [\begin {array}{c} \frac {1411 \left (-\frac {8 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}-\frac {16000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-23+16 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )\right )}{\left (\frac {167 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {334000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413-334 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )+1050 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}\right ) \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+27+\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )\right )} \\ -\frac {1600 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}+264 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}+\frac {1584000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-70703-1584 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{\frac {167 \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{3}+\frac {334000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}-47413-334 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )+1050 \left (-\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{60}-\frac {100}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}+\frac {18}{5}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}{6}-\frac {1000}{\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}}\right )}{10}\right )^{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Substitute in vector of dependent variables}\hspace {3pt} \\ {} & {} & \left [\begin {array}{c} x \left (t \right ) \\ y \\ z \left (t \right ) \end {array}\right ]=\left [\begin {array}{c} -\frac {10217142 \left (\left (\left (\frac {102037659}{567619}-\frac {209431 \,\mathrm {I} \sqrt {29760999}}{5108571}\right ) \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {2}{3}}+\left (\frac {79667}{1833}+\mathrm {I} \sqrt {29760999}\right ) \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}-\frac {624284032452}{567619}-\frac {413444644 \,\mathrm {I} \sqrt {29760999}}{1702857}\right ) c_{1} {\mathrm e}^{\frac {\left (\left (329940+60 \,\mathrm {I} \sqrt {3}\, \sqrt {9920333}\right )^{\frac {2}{3}}+108 \left (329940+60 \,\mathrm {I} \sqrt {3}\, \sqrt {9920333}\right )^{\frac {1}{3}}+6000\right ) t}{30 \left (329940+60 \,\mathrm {I} \sqrt {3}\, \sqrt {9920333}\right )^{\frac {1}{3}}}}+c_{2} \left (\left (-\frac {102037659}{1135238}-\frac {102037659 \,\mathrm {I} \sqrt {3}}{1135238}+\frac {209431 \,\mathrm {I} \sqrt {29760999}}{10217142}-\frac {209431 \sqrt {9920333}}{3405714}\right ) \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {2}{3}}+\left (\frac {79667}{1833}+\mathrm {I} \sqrt {29760999}\right ) \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}+\frac {206722322 \sqrt {9920333}}{567619}+\frac {312142016226}{567619}-\frac {312142016226 \,\mathrm {I} \sqrt {3}}{567619}+\frac {206722322 \,\mathrm {I} \sqrt {29760999}}{1702857}\right ) {\mathrm e}^{-\frac {2 \left (\sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )-3 \sqrt {5}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )-\frac {27}{5}\right ) t}{3}}+c_{3} \left (\left (-\frac {102037659}{1135238}+\frac {102037659 \,\mathrm {I} \sqrt {3}}{1135238}+\frac {209431 \,\mathrm {I} \sqrt {29760999}}{10217142}+\frac {209431 \sqrt {9920333}}{3405714}\right ) \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {2}{3}}+\left (\frac {79667}{1833}+\mathrm {I} \sqrt {29760999}\right ) \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}-\frac {206722322 \sqrt {9920333}}{567619}+\frac {312142016226}{567619}+\frac {312142016226 \,\mathrm {I} \sqrt {3}}{567619}+\frac {206722322 \,\mathrm {I} \sqrt {29760999}}{1702857}\right ) {\mathrm e}^{-\frac {2 \left (\sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )+3 \sqrt {5}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )-\frac {27}{5}\right ) t}{3}}\right ) \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {2}{3}}}{12058316711280 \,\mathrm {I} \sqrt {29760999}-64228778004671280} \\ \frac {\left (-2363719890 \,\mathrm {I}+200 \left (-30483501 \,\mathrm {I}+10999 \sqrt {29760999}\right ) \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}+\left (-55866337 \,\mathrm {I}+30163 \sqrt {29760999}\right ) \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {2}{3}}+54385110 \sqrt {29760999}\right ) c_{1} {\mathrm e}^{\frac {\left (\left (329940+60 \,\mathrm {I} \sqrt {3}\, \sqrt {9920333}\right )^{\frac {2}{3}}+108 \left (329940+60 \,\mathrm {I} \sqrt {3}\, \sqrt {9920333}\right )^{\frac {1}{3}}+6000\right ) t}{30 \left (329940+60 \,\mathrm {I} \sqrt {3}\, \sqrt {9920333}\right )^{\frac {1}{3}}}}+3048350100 c_{2} {\mathrm e}^{-\frac {2 \left (\sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )-3 \sqrt {5}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )-\frac {27}{5}\right ) t}{3}} \left (-\frac {78790663 \,\mathrm {I}}{101611670}+\left (-\sqrt {3}-\frac {10999 \sqrt {29760999}}{30483501}-\frac {10999 \,\mathrm {I} \sqrt {9920333}}{10161167}+\mathrm {I}\right ) \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}+\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {2}{3}} \left (\frac {55866337 \,\mathrm {I}}{3}+\frac {55866337 \sqrt {3}}{3}-\frac {30163 \sqrt {29760999}}{3}+30163 \,\mathrm {I} \sqrt {9920333}\right )}{2032233400}+\frac {1812837 \sqrt {29760999}}{101611670}\right )+3048350100 c_{3} {\mathrm e}^{-\frac {2 \left (\sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )+3 \sqrt {5}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )-\frac {27}{5}\right ) t}{3}} \left (-\frac {78790663 \,\mathrm {I}}{101611670}+\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}} \left (\sqrt {3}-\frac {10999 \sqrt {29760999}}{30483501}+\frac {10999 \,\mathrm {I} \sqrt {9920333}}{10161167}+\mathrm {I}\right )+\frac {\left (\frac {55866337 \,\mathrm {I}}{3}-\frac {55866337 \sqrt {3}}{3}-\frac {30163 \sqrt {29760999}}{3}-30163 \,\mathrm {I} \sqrt {9920333}\right ) \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {2}{3}}}{2032233400}+\frac {1812837 \sqrt {29760999}}{101611670}\right )}{98910 \left (1833 \sqrt {29760999}-79667 \,\mathrm {I}\right )} \\ {\mathrm e}^{\frac {\left (\left (329940+60 \,\mathrm {I} \sqrt {3}\, \sqrt {9920333}\right )^{\frac {2}{3}}+108 \left (329940+60 \,\mathrm {I} \sqrt {3}\, \sqrt {9920333}\right )^{\frac {1}{3}}+6000\right ) t}{30 \left (329940+60 \,\mathrm {I} \sqrt {3}\, \sqrt {9920333}\right )^{\frac {1}{3}}}} c_{1} +{\mathrm e}^{-\frac {2 \left (\sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )-3 \sqrt {5}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )-\frac {27}{5}\right ) t}{3}} c_{2} +{\mathrm e}^{-\frac {2 \left (\sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )+3 \sqrt {5}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )-\frac {27}{5}\right ) t}{3}} c_{3} \end {array}\right ] \\ \bullet & {} & \textrm {Solution to the system of ODEs}\hspace {3pt} \\ {} & {} & \left \{x \left (t \right )=-\frac {10217142 \left (\left (\left (\frac {102037659}{567619}-\frac {209431 \,\mathrm {I} \sqrt {29760999}}{5108571}\right ) \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {2}{3}}+\left (\frac {79667}{1833}+\mathrm {I} \sqrt {29760999}\right ) \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}-\frac {624284032452}{567619}-\frac {413444644 \,\mathrm {I} \sqrt {29760999}}{1702857}\right ) c_{1} {\mathrm e}^{\frac {\left (\left (329940+60 \,\mathrm {I} \sqrt {3}\, \sqrt {9920333}\right )^{\frac {2}{3}}+108 \left (329940+60 \,\mathrm {I} \sqrt {3}\, \sqrt {9920333}\right )^{\frac {1}{3}}+6000\right ) t}{30 \left (329940+60 \,\mathrm {I} \sqrt {3}\, \sqrt {9920333}\right )^{\frac {1}{3}}}}+c_{2} \left (\left (-\frac {102037659}{1135238}-\frac {102037659 \,\mathrm {I} \sqrt {3}}{1135238}+\frac {209431 \,\mathrm {I} \sqrt {29760999}}{10217142}-\frac {209431 \sqrt {9920333}}{3405714}\right ) \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {2}{3}}+\left (\frac {79667}{1833}+\mathrm {I} \sqrt {29760999}\right ) \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}+\frac {206722322 \sqrt {9920333}}{567619}+\frac {312142016226}{567619}-\frac {312142016226 \,\mathrm {I} \sqrt {3}}{567619}+\frac {206722322 \,\mathrm {I} \sqrt {29760999}}{1702857}\right ) {\mathrm e}^{-\frac {2 \left (\sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )-3 \sqrt {5}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )-\frac {27}{5}\right ) t}{3}}+c_{3} \left (\left (-\frac {102037659}{1135238}+\frac {102037659 \,\mathrm {I} \sqrt {3}}{1135238}+\frac {209431 \,\mathrm {I} \sqrt {29760999}}{10217142}+\frac {209431 \sqrt {9920333}}{3405714}\right ) \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {2}{3}}+\left (\frac {79667}{1833}+\mathrm {I} \sqrt {29760999}\right ) \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}-\frac {206722322 \sqrt {9920333}}{567619}+\frac {312142016226}{567619}+\frac {312142016226 \,\mathrm {I} \sqrt {3}}{567619}+\frac {206722322 \,\mathrm {I} \sqrt {29760999}}{1702857}\right ) {\mathrm e}^{-\frac {2 \left (\sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )+3 \sqrt {5}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )-\frac {27}{5}\right ) t}{3}}\right ) \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {2}{3}}}{12058316711280 \,\mathrm {I} \sqrt {29760999}-64228778004671280}, y=\frac {\left (-2363719890 \,\mathrm {I}+200 \left (-30483501 \,\mathrm {I}+10999 \sqrt {29760999}\right ) \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}+\left (-55866337 \,\mathrm {I}+30163 \sqrt {29760999}\right ) \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {2}{3}}+54385110 \sqrt {29760999}\right ) c_{1} {\mathrm e}^{\frac {\left (\left (329940+60 \,\mathrm {I} \sqrt {3}\, \sqrt {9920333}\right )^{\frac {2}{3}}+108 \left (329940+60 \,\mathrm {I} \sqrt {3}\, \sqrt {9920333}\right )^{\frac {1}{3}}+6000\right ) t}{30 \left (329940+60 \,\mathrm {I} \sqrt {3}\, \sqrt {9920333}\right )^{\frac {1}{3}}}}+3048350100 c_{2} {\mathrm e}^{-\frac {2 \left (\sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )-3 \sqrt {5}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )-\frac {27}{5}\right ) t}{3}} \left (-\frac {78790663 \,\mathrm {I}}{101611670}+\left (-\sqrt {3}-\frac {10999 \sqrt {29760999}}{30483501}-\frac {10999 \,\mathrm {I} \sqrt {9920333}}{10161167}+\mathrm {I}\right ) \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}}+\frac {\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {2}{3}} \left (\frac {55866337 \,\mathrm {I}}{3}+\frac {55866337 \sqrt {3}}{3}-\frac {30163 \sqrt {29760999}}{3}+30163 \,\mathrm {I} \sqrt {9920333}\right )}{2032233400}+\frac {1812837 \sqrt {29760999}}{101611670}\right )+3048350100 c_{3} {\mathrm e}^{-\frac {2 \left (\sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )+3 \sqrt {5}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )-\frac {27}{5}\right ) t}{3}} \left (-\frac {78790663 \,\mathrm {I}}{101611670}+\left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {1}{3}} \left (\sqrt {3}-\frac {10999 \sqrt {29760999}}{30483501}+\frac {10999 \,\mathrm {I} \sqrt {9920333}}{10161167}+\mathrm {I}\right )+\frac {\left (\frac {55866337 \,\mathrm {I}}{3}-\frac {55866337 \sqrt {3}}{3}-\frac {30163 \sqrt {29760999}}{3}-30163 \,\mathrm {I} \sqrt {9920333}\right ) \left (329940+60 \,\mathrm {I} \sqrt {29760999}\right )^{\frac {2}{3}}}{2032233400}+\frac {1812837 \sqrt {29760999}}{101611670}\right )}{98910 \left (1833 \sqrt {29760999}-79667 \,\mathrm {I}\right )}, z \left (t \right )={\mathrm e}^{\frac {\left (\left (329940+60 \,\mathrm {I} \sqrt {3}\, \sqrt {9920333}\right )^{\frac {2}{3}}+108 \left (329940+60 \,\mathrm {I} \sqrt {3}\, \sqrt {9920333}\right )^{\frac {1}{3}}+6000\right ) t}{30 \left (329940+60 \,\mathrm {I} \sqrt {3}\, \sqrt {9920333}\right )^{\frac {1}{3}}}} c_{1} +{\mathrm e}^{-\frac {2 \left (\sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )-3 \sqrt {5}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )-\frac {27}{5}\right ) t}{3}} c_{2} +{\mathrm e}^{-\frac {2 \left (\sqrt {3}\, \sqrt {5}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )+3 \sqrt {5}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {29760999}}{5499}\right )}{3}\right )-\frac {27}{5}\right ) t}{3}} c_{3} \right \} \end {array} \]
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 1014
dsolve([diff(x(t),t)=9/10*x(t)+21/10*y(t)+32/10*z(t),diff(y(t),t)=7/10*x(t)+65/10*y(t)+42/10*z(t),diff(z(t),t)=11/10*x(t)+17/10*y(t)+34/10*z(t)],singsol=all)
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{-\frac {\left (i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}+\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-216 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}+c_{2} {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}-\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+216 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}+c_{3} {\mathrm e}^{\frac {\left (\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+108 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right ) t}{30 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}} \\ y \left (t \right ) &= \frac {\left (8 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}-8 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+207000 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-299382930 i \sqrt {3}+20757 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-2070 i \sqrt {29760999}+6210 \sqrt {9920333}-207000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-299382930\right ) c_{1} {\mathrm e}^{-\frac {\left (i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}+\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-216 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}}{88731 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}-\frac {\left (8 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+8 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+207000 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-299382930 i \sqrt {3}-20757 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+2070 i \sqrt {29760999}+207000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6210 \sqrt {9920333}+299382930\right ) c_{2} {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}-\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+216 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}}{88731 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}+\frac {\left (16 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+20757 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+4140 i \sqrt {29760999}+414000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+598765860\right ) c_{3} {\mathrm e}^{\frac {\left (\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+108 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right ) t}{30 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}}{88731 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}} \\ z \left (t \right ) &= -\frac {\left (7 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}-7 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}-3516000 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-58655160 i \sqrt {3}-81660 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+35160 i \sqrt {29760999}+3516000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-105480 \sqrt {9920333}-58655160\right ) c_{1} {\mathrm e}^{-\frac {\left (i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}+\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-216 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}}{118308 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}+\frac {\left (7 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}+7 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}-3516000 i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-58655160 i \sqrt {3}+81660 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-35160 i \sqrt {29760999}-3516000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-105480 \sqrt {9920333}+58655160\right ) c_{2} {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-6000 i \sqrt {3}-\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+216 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}-6000\right ) t}{60 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}}{118308 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}}-\frac {\left (7 \left (329940+60 i \sqrt {29760999}\right )^{\frac {4}{3}}-40830 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}-35160 i \sqrt {29760999}-3516000 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+58655160\right ) c_{3} {\mathrm e}^{\frac {\left (\left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}+108 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}+6000\right ) t}{30 \left (329940+60 i \sqrt {29760999}\right )^{\frac {1}{3}}}}}{59154 \left (329940+60 i \sqrt {29760999}\right )^{\frac {2}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.029 (sec). Leaf size: 616
DSolve[{x'[t]==9/10*x[t]+21/10*y[t]+32/10*z[t],y'[t]==7/10*x[t]+65/10*y[t]+42/10*z[t],z'[t]==11/10*x[t]+17/10*y[t]+34/10*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to 2 c_3 \text {RootSum}\left [\text {$\#$1}^3-108 \text {$\#$1}^2+1888 \text {$\#$1}+904\&,\frac {16 \text {$\#$1} e^{\frac {\text {$\#$1} t}{10}}-599 e^{\frac {\text {$\#$1} t}{10}}}{3 \text {$\#$1}^2-216 \text {$\#$1}+1888}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3-108 \text {$\#$1}^2+1888 \text {$\#$1}+904\&,\frac {21 \text {$\#$1} e^{\frac {\text {$\#$1} t}{10}}-170 e^{\frac {\text {$\#$1} t}{10}}}{3 \text {$\#$1}^2-216 \text {$\#$1}+1888}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3-108 \text {$\#$1}^2+1888 \text {$\#$1}+904\&,\frac {\text {$\#$1}^2 e^{\frac {\text {$\#$1} t}{10}}-99 \text {$\#$1} e^{\frac {\text {$\#$1} t}{10}}+1496 e^{\frac {\text {$\#$1} t}{10}}}{3 \text {$\#$1}^2-216 \text {$\#$1}+1888}\&\right ] \\ y(t)\to 7 c_1 \text {RootSum}\left [\text {$\#$1}^3-108 \text {$\#$1}^2+1888 \text {$\#$1}+904\&,\frac {\text {$\#$1} e^{\frac {\text {$\#$1} t}{10}}+32 e^{\frac {\text {$\#$1} t}{10}}}{3 \text {$\#$1}^2-216 \text {$\#$1}+1888}\&\right ]+14 c_3 \text {RootSum}\left [\text {$\#$1}^3-108 \text {$\#$1}^2+1888 \text {$\#$1}+904\&,\frac {3 \text {$\#$1} e^{\frac {\text {$\#$1} t}{10}}-11 e^{\frac {\text {$\#$1} t}{10}}}{3 \text {$\#$1}^2-216 \text {$\#$1}+1888}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3-108 \text {$\#$1}^2+1888 \text {$\#$1}+904\&,\frac {\text {$\#$1}^2 e^{\frac {\text {$\#$1} t}{10}}-43 \text {$\#$1} e^{\frac {\text {$\#$1} t}{10}}-46 e^{\frac {\text {$\#$1} t}{10}}}{3 \text {$\#$1}^2-216 \text {$\#$1}+1888}\&\right ] \\ z(t)\to c_1 \text {RootSum}\left [\text {$\#$1}^3-108 \text {$\#$1}^2+1888 \text {$\#$1}+904\&,\frac {11 \text {$\#$1} e^{\frac {\text {$\#$1} t}{10}}-596 e^{\frac {\text {$\#$1} t}{10}}}{3 \text {$\#$1}^2-216 \text {$\#$1}+1888}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3-108 \text {$\#$1}^2+1888 \text {$\#$1}+904\&,\frac {17 \text {$\#$1} e^{\frac {\text {$\#$1} t}{10}}+78 e^{\frac {\text {$\#$1} t}{10}}}{3 \text {$\#$1}^2-216 \text {$\#$1}+1888}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3-108 \text {$\#$1}^2+1888 \text {$\#$1}+904\&,\frac {\text {$\#$1}^2 e^{\frac {\text {$\#$1} t}{10}}-74 \text {$\#$1} e^{\frac {\text {$\#$1} t}{10}}+438 e^{\frac {\text {$\#$1} t}{10}}}{3 \text {$\#$1}^2-216 \text {$\#$1}+1888}\&\right ] \\ \end{align*}