Internal problem ID [6719]
Internal file name [OUTPUT/5967_Sunday_June_05_2022_04_08_12_PM_40291562/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 8.1. Page 332
Problem number: 9.
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 )&=x \left (t \right )-y \left (t \right )+2 z \left (t \right )+{\mathrm e}^{-t}-3 t\\ y^{\prime }\left (t \right )&=3 x \left (t \right )-4 y \left (t \right )+z \left (t \right )+2 \,{\mathrm e}^{-t}+t\\ z^{\prime }\left (t \right )&=-2 x \left (t \right )+5 y \left (t \right )+6 z \left (t \right )+2 \,{\mathrm e}^{-t}-t \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) + \vec {G}(t) \end {align*}
Or \begin {align*} \left [\begin {array}{c} x^{\prime }\left (t \right ) \\ y^{\prime }\left (t \right ) \\ z^{\prime }\left (t \right ) \end {array}\right ] &= \left [\begin {array}{ccc} 1 & -1 & 2 \\ 3 & -4 & 1 \\ -2 & 5 & 6 \end {array}\right ]\, \left [\begin {array}{c} x \left (t \right ) \\ y \left (t \right ) \\ z \left (t \right ) \end {array}\right ] + \left [\begin {array}{c} {\mathrm e}^{-t}-3 t \\ 2 \,{\mathrm e}^{-t}+t \\ 2 \,{\mathrm e}^{-t}-t \end {array}\right ] \end {align*}
Since the system is nonhomogeneous, then the solution is given by \begin {align*} \vec {x}(t) &= \vec {x}_h(t) + \vec {x}_p(t) \end {align*}
Where \(\vec {x}_h(t)\) is the homogeneous solution to \(\vec {x}'(t) = A\, \vec {x}(t)\) and \(\vec {x}_p(t)\) is a particular solution to \(\vec {x}'(t) = A\, \vec {x}(t) + \vec {G}(t)\). The particular solution will be found using variation of parameters method applied to the fundamental matrix.
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} 1 & -1 & 2 \\ 3 & -4 & 1 \\ -2 & 5 & 6 \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} 1-\lambda & -1 & 2 \\ 3 & -4-\lambda & 1 \\ -2 & 5 & 6-\lambda \end {array}\right ]\right ) &= 0 \end {align*}
Which gives the characteristic equation \begin {align*} \lambda ^{3}-3 \lambda ^{2}-20 \lambda -5&=0 \end {align*}
The roots of the above are the eigenvalues. \begin {align*} \lambda _1 &= \frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1\\ \lambda _2 &= -\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\\ \lambda _3 &= -\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}
This table summarises the above result
| eigenvalue | algebraic multiplicity | type of eigenvalue |
| \(-\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\) | \(1\) | complex eigenvalue |
| \(-\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\) | \(1\) | complex eigenvalue |
| \(\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1\) | \(1\) | complex eigenvalue |
Now the eigenvector for each eigenvalue are found.
Considering the eigenvalue \(\lambda _{1} = \frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1\)
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} 1 & -1 & 2 \\ 3 & -4 & 1 \\ -2 & 5 & 6 \end {array}\right ] - \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1\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 (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276}{6 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}} & -1 & 2 \\ 3 & \frac {-\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}-30 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-276}{6 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}} & 1 \\ -2 & 5 & \frac {-\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+30 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-276}{6 \left (2916+12 i \sqrt {86955}\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 \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}&-1&2&0\\ 3&-5-\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}&1&0\\ -2&5&5-\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}-\frac {3 R_{1}}{-\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276}{6 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}&-1&2&0\\ 0&\frac {\left (-2 i \sqrt {86955}-1866\right ) \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-110 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}-60 i \sqrt {86955}-27276}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \left (\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276\right )}&\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+36 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+276}{\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276}&0\\ -2&5&\frac {-\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+30 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-276}{6 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}-\frac {12 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} R_{1}}{\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276}{6 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}&-1&2&0\\ 0&\frac {\left (-2 i \sqrt {86955}-1866\right ) \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-110 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}-60 i \sqrt {86955}-27276}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \left (\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276\right )}&\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+36 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+276}{\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276}&0\\ 0&\frac {5 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+12 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+1380}{\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276}&\frac {\left (-2 i \sqrt {86955}+894\right ) \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-116 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+60 i \sqrt {86955}+1884}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \left (\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276\right )}&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}-\frac {\left (5 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+12 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+1380\right ) \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} R_{2}}{\left (-2 i \sqrt {86955}-1866\right ) \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-110 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}-60 i \sqrt {86955}-27276} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276}{6 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}&-1&2&0\\ 0&\frac {\left (-2 i \sqrt {86955}-1866\right ) \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-110 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}-60 i \sqrt {86955}-27276}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \left (\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276\right )}&\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+36 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+276}{\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276}&0\\ 0&0&0&0 \end {array} \right ] \end {align*}
Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} -\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276}{6 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}} & -1 & 2 \\ 0 & \frac {\left (-2 i \sqrt {86955}-1866\right ) \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-110 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}-60 i \sqrt {86955}-27276}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \left (\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276\right )} & \frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+36 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+276}{\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276} \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] 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 \(\left \{v_{1} = \frac {12 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} t \left (i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+27 i \sqrt {86955}+46 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+864 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+12909\right )}{\left (i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+30 i \sqrt {86955}+55 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+933 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+13638\right ) \left (\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276\right )}, v_{2} = \frac {6 t \left (i \sqrt {86955}+3 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+23 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+243\right )}{i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+30 i \sqrt {86955}+55 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+933 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+13638}\right \}\)
Hence the solution is \[ \left [\begin {array}{c} v_{1} \\ v_{2} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {12 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} t \left (i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+27 i \sqrt {86955}+46 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+864 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+12909\right )}{\left (i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+30 i \sqrt {86955}+55 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+933 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+13638\right ) \left (\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276\right )} \\ \frac {6 t \left (i \sqrt {86955}+3 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+23 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+243\right )}{i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+30 i \sqrt {86955}+55 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+933 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+13638} \\ t \end {array}\right ] \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \left [\begin {array}{c} v_{1} \\ v_{2} \\ t \end {array}\right ] = t \left [\begin {array}{c} \frac {12 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \left (i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+27 i \sqrt {86955}+46 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+864 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+12909\right )}{\left (i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+30 i \sqrt {86955}+55 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+933 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+13638\right ) \left (\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276\right )} \\ \frac {6 i \sqrt {86955}+18 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+138 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+1458}{i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+30 i \sqrt {86955}+55 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+933 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+13638} \\ 1 \end {array}\right ] \] Let \(t = 1\) the eigenvector becomes \[ \left [\begin {array}{c} v_{1} \\ v_{2} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {12 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \left (i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+27 i \sqrt {86955}+46 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+864 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+12909\right )}{\left (i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+30 i \sqrt {86955}+55 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+933 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+13638\right ) \left (\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276\right )} \\ \frac {6 i \sqrt {86955}+18 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+138 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+1458}{i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+30 i \sqrt {86955}+55 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+933 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+13638} \\ 1 \end {array}\right ] \] Which is normalized to \[ \left [\begin {array}{c} v_{1} \\ v_{2} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {12 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \left (i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+27 i \sqrt {86955}+46 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+864 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+12909\right )}{\left (i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+30 i \sqrt {86955}+55 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+933 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+13638\right ) \left (\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276\right )} \\ \frac {6 i \sqrt {86955}+18 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+138 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+1458}{i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+30 i \sqrt {86955}+55 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+933 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+13638} \\ 1 \end {array}\right ] \] Considering the eigenvalue \(\lambda _{2} = -\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\)
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} 1 & -1 & 2 \\ 3 & -4 & 1 \\ -2 & 5 & 6 \end {array}\right ] - \left (-\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\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 {\sqrt {23}\, \left (\sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )-3 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )\right )}{3} & -1 & 2 \\ 3 & -5+\frac {\sqrt {3}\, \sqrt {23}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )}{3}-\sqrt {23}\, \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right ) & 1 \\ -2 & 5 & 5+\frac {\sqrt {3}\, \sqrt {23}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )}{3}-\sqrt {23}\, \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right ) \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 \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}+\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}&-1&2&0\\ 3&-5+\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}+\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}&1&0\\ -2&5&5+\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}+\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}-\frac {3 R_{1}}{\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}+\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {\sqrt {23}\, \left (\sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )-3 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )\right )}{3}&-1&2&0\\ 0&\frac {\left (-6 i \sqrt {28985}-1866 i-1866 \sqrt {3}+2 \sqrt {86955}\right ) \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+220 i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+180 i \sqrt {28985}-27276 i+27276 \sqrt {3}+60 \sqrt {86955}}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \left (\left (276-\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}\right ) \sqrt {3}+i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276 i\right )}&\frac {i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}-276 i \sqrt {3}+\left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-72 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+276}{i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}-276 i \sqrt {3}+\left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+276}&0\\ -2&5&5+\frac {\sqrt {3}\, \sqrt {23}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )}{3}-\sqrt {23}\, \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}+\frac {6 \sqrt {23}\, R_{1}}{23 \left (\sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )-3 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {\sqrt {23}\, \left (\sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )-3 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )\right )}{3}&-1&2&0\\ 0&\frac {\left (-6 i \sqrt {28985}-1866 i-1866 \sqrt {3}+2 \sqrt {86955}\right ) \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+220 i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+180 i \sqrt {28985}-27276 i+27276 \sqrt {3}+60 \sqrt {86955}}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \left (\left (276-\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}\right ) \sqrt {3}+i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276 i\right )}&\frac {i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}-276 i \sqrt {3}+\left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-72 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+276}{i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}-276 i \sqrt {3}+\left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+276}&0\\ 0&\frac {115 \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )-345 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )-6 \sqrt {23}}{23 \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )-69 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )}&\frac {69 \sqrt {23}\, \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )^{2}+\left (-46 \sqrt {3}\, \sqrt {23}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )-345\right ) \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )+23 \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )^{2} \sqrt {23}+115 \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )+12 \sqrt {23}}{23 \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )-69 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )}&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}-\frac {\left (115 \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )-345 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )-6 \sqrt {23}\right ) \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \left (\left (276-\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}\right ) \sqrt {3}+i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276 i\right ) R_{2}}{\left (23 \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )-69 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )\right ) \left (\left (-6 i \sqrt {28985}-1866 i-1866 \sqrt {3}+2 \sqrt {86955}\right ) \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+220 i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+180 i \sqrt {28985}-27276 i+27276 \sqrt {3}+60 \sqrt {86955}\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {\sqrt {23}\, \left (\sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )-3 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )\right )}{3}&-1&2&0\\ 0&\frac {\left (-6 i \sqrt {28985}-1866 i-1866 \sqrt {3}+2 \sqrt {86955}\right ) \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+220 i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+180 i \sqrt {28985}-27276 i+27276 \sqrt {3}+60 \sqrt {86955}}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \left (\left (276-\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}\right ) \sqrt {3}+i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276 i\right )}&\frac {i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}-276 i \sqrt {3}+\left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-72 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+276}{i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}-276 i \sqrt {3}+\left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+276}&0\\ 0&0&\frac {5 \left (12 \left (3531+8 \left (1+i \sqrt {3}\right ) \sqrt {28985}-1177 i \sqrt {3}\right ) \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+\left (-2937+3 \left (1-i \sqrt {3}\right ) \sqrt {28985}-979 i \sqrt {3}\right ) \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-4284 \sqrt {28985}+347004 i \sqrt {3}\right ) \sin \left (\frac {\arctan \left (\frac {81 \sqrt {3}\, \sqrt {28985}}{4651}\right )}{6}\right )+26 \sqrt {23}\, \left (\frac {4316166 i}{299}+\cos \left (\frac {\arctan \left (\frac {81 \sqrt {3}\, \sqrt {28985}}{4651}\right )}{3}\right ) \left (-\frac {151146 i}{13}+\left (\frac {5804 i}{13}-\sqrt {28985}\, \left (\sqrt {3}+3 i\right )-\frac {5804 \sqrt {3}}{13}\right ) \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+\frac {\left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \left (1601 i+5 \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {28985}+1601 \sqrt {3}\right )}{52}+\frac {622 \sqrt {3}\, \sqrt {28985}}{13}\right )+\frac {2 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \left (-3577 i+8 \sqrt {28985}\, \left (\sqrt {3}+3 i\right )+3577 \sqrt {3}\right )}{13}-\frac {\left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \left (22679 i+71 \left (i-\frac {\sqrt {3}}{3}\right ) \sqrt {28985}+22679 \sqrt {3}\right )}{598}-\frac {17762 \sqrt {3}\, \sqrt {28985}}{299}\right )}{78 \left (\left (\frac {5804}{13}+\left (1+i \sqrt {3}\right ) \sqrt {28985}-\frac {5804 i \sqrt {3}}{39}\right ) \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+\frac {\left (-1601+\frac {5 \left (1-i \sqrt {3}\right ) \sqrt {28985}}{3}-\frac {1601 i \sqrt {3}}{3}\right ) \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}}{52}-\frac {622 \sqrt {28985}}{13}+\frac {50382 i \sqrt {3}}{13}\right ) \sin \left (\frac {\arctan \left (\frac {81 \sqrt {3}\, \sqrt {28985}}{4651}\right )}{6}\right )}&0 \end {array} \right ] \end {align*}
Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} \frac {\sqrt {23}\, \left (\sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )-3 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )\right )}{3} & -1 & 2 \\ 0 & \frac {\left (-6 i \sqrt {28985}-1866 i-1866 \sqrt {3}+2 \sqrt {86955}\right ) \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+220 i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+180 i \sqrt {28985}-27276 i+27276 \sqrt {3}+60 \sqrt {86955}}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \left (\left (276-\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}\right ) \sqrt {3}+i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276 i\right )} & \frac {i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}-276 i \sqrt {3}+\left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-72 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+276}{i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}-276 i \sqrt {3}+\left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+276} \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] 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 \sqrt {23}\, \left (9 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {86955}+9 i \sqrt {28985}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+2845 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {3}-276 \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-8535 i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276 i \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \sqrt {28985}+41400 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \sqrt {3}+124200 i \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-13824 i \sqrt {28985}-1119744 \sqrt {3}\right ) t \csc \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}\right )}{46 \left (-468 \sqrt {28985}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+156 i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+5 i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+15 \sqrt {28985}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}-4803 i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {3}+4803 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}-7464 i \sqrt {86955}+69648 i \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \sqrt {3}+69648 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-1813752\right )}, v_{2} = \frac {t \left (3 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+i \sqrt {28985}\, \sqrt {3}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-1104 i \sqrt {3}\, \sqrt {28985}-108 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+36 i \sqrt {3}\, \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+1071 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-1071 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}+15096 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+15096 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \sqrt {3}-268272\right )}{-468 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+156 i \sqrt {3}\, \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+5 i \sqrt {28985}\, \sqrt {3}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+15 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-4803 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}+4803 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-7464 i \sqrt {3}\, \sqrt {28985}+69648 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \sqrt {3}+69648 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}-1813752}\right \}\)
Hence the solution is \[ \left [\begin {array}{c} v_{1} \\ v_{2} \\ t \end {array}\right ] = \left [\begin {array}{c} -\frac {3 \sqrt {23}\, \left (9 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {86955}+9 i \sqrt {28985}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+2845 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {3}-276 \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-8535 i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276 i \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \sqrt {28985}+41400 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \sqrt {3}+124200 i \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-13824 i \sqrt {28985}-1119744 \sqrt {3}\right ) t \csc \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}\right )}{46 \left (-468 \sqrt {28985}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+156 i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+5 i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+15 \sqrt {28985}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}-4803 i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {3}+4803 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}-7464 i \sqrt {86955}+69648 i \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \sqrt {3}+69648 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-1813752\right )} \\ \frac {t \left (3 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+i \sqrt {28985}\, \sqrt {3}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-1104 i \sqrt {3}\, \sqrt {28985}-108 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+36 i \sqrt {3}\, \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+1071 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-1071 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}+15096 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+15096 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \sqrt {3}-268272\right )}{-468 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+156 i \sqrt {3}\, \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+5 i \sqrt {28985}\, \sqrt {3}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+15 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-4803 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}+4803 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-7464 i \sqrt {3}\, \sqrt {28985}+69648 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \sqrt {3}+69648 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}-1813752} \\ 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} v_{1} \\ v_{2} \\ t \end {array}\right ] = t \left [\begin {array}{c} -\frac {3 \sqrt {23}\, \left (9 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {86955}+9 i \sqrt {28985}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+2845 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {3}-276 \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-8535 i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276 i \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \sqrt {28985}+41400 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \sqrt {3}+124200 i \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-13824 i \sqrt {28985}-1119744 \sqrt {3}\right ) \csc \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}\right )}{46 \left (-468 \sqrt {28985}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+156 i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+5 i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+15 \sqrt {28985}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}-4803 i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {3}+4803 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}-7464 i \sqrt {86955}+69648 i \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \sqrt {3}+69648 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-1813752\right )} \\ \frac {3 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+i \sqrt {28985}\, \sqrt {3}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-1104 i \sqrt {3}\, \sqrt {28985}-108 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+36 i \sqrt {3}\, \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+1071 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-1071 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}+15096 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+15096 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \sqrt {3}-268272}{-468 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+156 i \sqrt {3}\, \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+5 i \sqrt {28985}\, \sqrt {3}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+15 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-4803 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}+4803 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-7464 i \sqrt {3}\, \sqrt {28985}+69648 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \sqrt {3}+69648 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}-1813752} \\ 1 \end {array}\right ] \] Let \(t = 1\) the eigenvector becomes \[ \left [\begin {array}{c} v_{1} \\ v_{2} \\ t \end {array}\right ] = \left [\begin {array}{c} -\frac {3 \sqrt {23}\, \left (9 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {86955}+9 i \sqrt {28985}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+2845 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {3}-276 \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-8535 i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276 i \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \sqrt {28985}+41400 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \sqrt {3}+124200 i \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-13824 i \sqrt {28985}-1119744 \sqrt {3}\right ) \csc \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}\right )}{46 \left (-468 \sqrt {28985}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+156 i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+5 i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+15 \sqrt {28985}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}-4803 i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {3}+4803 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}-7464 i \sqrt {86955}+69648 i \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \sqrt {3}+69648 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-1813752\right )} \\ \frac {3 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+i \sqrt {28985}\, \sqrt {3}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-1104 i \sqrt {3}\, \sqrt {28985}-108 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+36 i \sqrt {3}\, \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+1071 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-1071 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}+15096 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+15096 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \sqrt {3}-268272}{-468 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+156 i \sqrt {3}\, \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+5 i \sqrt {28985}\, \sqrt {3}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+15 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-4803 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}+4803 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-7464 i \sqrt {3}\, \sqrt {28985}+69648 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \sqrt {3}+69648 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}-1813752} \\ 1 \end {array}\right ] \] Which is normalized to \[ \left [\begin {array}{c} v_{1} \\ v_{2} \\ t \end {array}\right ] = \left [\begin {array}{c} -\frac {3 \sqrt {23}\, \left (9 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {86955}+9 i \sqrt {28985}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+2845 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {3}-276 \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-8535 i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+276 i \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \sqrt {28985}+41400 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \sqrt {3}+124200 i \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-13824 i \sqrt {28985}-1119744 \sqrt {3}\right ) \csc \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}\right )}{46 \left (-468 \sqrt {28985}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+156 i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+5 i \sqrt {86955}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+15 \sqrt {28985}\, \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}-4803 i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {3}+4803 \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}-7464 i \sqrt {86955}+69648 i \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}} \sqrt {3}+69648 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-1813752\right )} \\ \frac {3 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+i \sqrt {28985}\, \sqrt {3}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-1104 i \sqrt {3}\, \sqrt {28985}-108 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+36 i \sqrt {3}\, \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+1071 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-1071 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}+15096 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+15096 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \sqrt {3}-268272}{-468 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+156 i \sqrt {3}\, \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+5 i \sqrt {28985}\, \sqrt {3}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+15 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-4803 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}+4803 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-7464 i \sqrt {3}\, \sqrt {28985}+69648 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \sqrt {3}+69648 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}-1813752} \\ 1 \end {array}\right ] \] Considering the eigenvalue \(\lambda _{3} = -\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\)
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} 1 & -1 & 2 \\ 3 & -4 & 1 \\ -2 & 5 & 6 \end {array}\right ] - \left (-\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\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 {\sqrt {23}\, \left (\sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )+3 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )\right )}{3} & -1 & 2 \\ 3 & -5+\frac {\sqrt {3}\, \sqrt {23}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )}{3}+\sqrt {23}\, \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right ) & 1 \\ -2 & 5 & 5+\frac {\sqrt {3}\, \sqrt {23}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )}{3}+\sqrt {23}\, \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right ) \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 \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}+\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}&-1&2&0\\ 3&-5+\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}+\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}&1&0\\ -2&5&5+\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}+\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}-\frac {3 R_{1}}{\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}+\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {\sqrt {23}\, \left (\sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )+3 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )\right )}{3}&-1&2&0\\ 0&\frac {27276+2 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \left (933+i \left (933+\sqrt {28985}\right ) \sqrt {3}-3 \sqrt {28985}\right )-220 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+12 i \left (-2273+5 \sqrt {28985}\right ) \sqrt {3}+180 \sqrt {28985}}{\left (i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}-276 i \sqrt {3}-\left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-276\right ) \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}}&\frac {i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {3}-276 i \sqrt {3}-\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+72 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-276}{i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {3}-276 i \sqrt {3}-\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}-276}&0\\ -2&5&5+\frac {\sqrt {3}\, \sqrt {23}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )}{3}+\sqrt {23}\, \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}+\frac {6 \sqrt {23}\, R_{1}}{23 \left (\sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )+3 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {\sqrt {23}\, \left (\sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )+3 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )\right )}{3}&-1&2&0\\ 0&\frac {27276+2 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \left (933+i \left (933+\sqrt {28985}\right ) \sqrt {3}-3 \sqrt {28985}\right )-220 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+12 i \left (-2273+5 \sqrt {28985}\right ) \sqrt {3}+180 \sqrt {28985}}{\left (i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}-276 i \sqrt {3}-\left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-276\right ) \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}}&\frac {i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {3}-276 i \sqrt {3}-\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+72 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-276}{i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {3}-276 i \sqrt {3}-\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}-276}&0\\ 0&\frac {115 \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )+345 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )-6 \sqrt {23}}{23 \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )+69 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )}&\frac {69 \sqrt {23}\, \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )^{2}+\left (46 \sqrt {3}\, \sqrt {23}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )+345\right ) \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )+23 \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )^{2} \sqrt {23}+115 \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )+12 \sqrt {23}}{23 \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )+69 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )}&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}-\frac {\left (115 \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )+345 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )-6 \sqrt {23}\right ) \left (i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}-276 i \sqrt {3}-\left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-276\right ) \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} R_{2}}{2 \left (23 \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )+69 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )\right ) \left (13638+\left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \left (933+i \left (933+\sqrt {28985}\right ) \sqrt {3}-3 \sqrt {28985}\right )-110 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+6 i \left (-2273+5 \sqrt {28985}\right ) \sqrt {3}+90 \sqrt {28985}\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {\sqrt {23}\, \left (\sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )+3 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )\right )}{3}&-1&2&0\\ 0&\frac {27276+2 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \left (933+i \left (933+\sqrt {28985}\right ) \sqrt {3}-3 \sqrt {28985}\right )-220 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+12 i \left (-2273+5 \sqrt {28985}\right ) \sqrt {3}+180 \sqrt {28985}}{\left (i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}-276 i \sqrt {3}-\left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-276\right ) \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}}&\frac {i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {3}-276 i \sqrt {3}-\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+72 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-276}{i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {3}-276 i \sqrt {3}-\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}-276}&0\\ 0&0&\frac {15 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right ) \left (15096+\left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \left (1071+i \left (\sqrt {28985}+1071\right ) \sqrt {3}-3 \sqrt {28985}\right )-146 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+12 i \left (3 \sqrt {28985}-1258\right ) \sqrt {3}+108 \sqrt {28985}\right )+5 \left (-45288 i+3 \left (1071 i+\left (-\sqrt {28985}+357\right ) \sqrt {3}+i \sqrt {28985}\right ) \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}-146 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}+12 \left (1258+9 \sqrt {28985}\right ) \sqrt {3}+108 i \sqrt {28985}\right ) \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )+\left (\frac {782256}{23}+\left (-40914 i+3 \left (933 i-\sqrt {3}\, \left (\sqrt {28985}-311\right )+i \sqrt {28985}\right ) \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}-110 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}-6 \left (-2273-15 \sqrt {28985}\right ) \sqrt {3}+90 i \sqrt {28985}\right ) \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{3}+\frac {\pi }{6}\right )+\left (-13638+\left (-933-i \left (933+\sqrt {28985}\right ) \sqrt {3}+3 \sqrt {28985}\right ) \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+110 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-6 i \left (-2273+5 \sqrt {28985}\right ) \sqrt {3}-90 \sqrt {28985}\right ) \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{3}+\frac {\pi }{6}\right )+\frac {2 \left (26643-i \left (-26643-29 \sqrt {28985}\right ) \sqrt {3}-87 \sqrt {28985}\right ) \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}}{23}-268 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-\frac {24 i \left (-71 \sqrt {28985}+32594\right ) \sqrt {3}}{23}+\frac {5112 \sqrt {28985}}{23}\right ) \sqrt {23}}{3 \left (\frac {\sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )}{3}+\cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )\right ) \left (13638+\left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \left (933+i \left (933+\sqrt {28985}\right ) \sqrt {3}-3 \sqrt {28985}\right )-110 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+6 i \left (-2273+5 \sqrt {28985}\right ) \sqrt {3}+90 \sqrt {28985}\right )}&0 \end {array} \right ] \end {align*}
Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} \frac {\sqrt {23}\, \left (\sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )+3 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}+\frac {\pi }{3}\right )\right )}{3} & -1 & 2 \\ 0 & \frac {27276+2 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \left (933+i \left (933+\sqrt {28985}\right ) \sqrt {3}-3 \sqrt {28985}\right )-220 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+12 i \left (-2273+5 \sqrt {28985}\right ) \sqrt {3}+180 \sqrt {28985}}{\left (i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}-276 i \sqrt {3}-\left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}-276\right ) \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}} & \frac {i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {3}-276 i \sqrt {3}-\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}+72 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}-276}{i \left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}} \sqrt {3}-276 i \sqrt {3}-\left (2916+12 i \sqrt {86955}\right )^{\frac {2}{3}}-276} \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] 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 {9 \sqrt {23}\, \left (3 i \sqrt {28985}\, \sqrt {3}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+92 i \sqrt {3}\, \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+2845 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}-4608 i \sqrt {3}\, \sqrt {28985}-41400 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \sqrt {3}-9 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+276 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+2845 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+41400 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}-1119744\right ) t}{23 \left (\sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}\right )-3 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}\right )\right ) \left (5 i \sqrt {28985}\, \sqrt {3}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+156 i \sqrt {3}\, \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+4803 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}-7464 i \sqrt {3}\, \sqrt {28985}-69648 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \sqrt {3}-15 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+468 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+4803 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+69648 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}-1813752\right )}, v_{2} = \frac {\left (i \sqrt {28985}\, \sqrt {3}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+36 i \sqrt {3}\, \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+1071 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}-1104 i \sqrt {3}\, \sqrt {28985}-15096 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \sqrt {3}-3 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+108 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+1071 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+15096 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}-268272\right ) t}{5 i \sqrt {28985}\, \sqrt {3}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+156 i \sqrt {3}\, \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+4803 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}-7464 i \sqrt {3}\, \sqrt {28985}-69648 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \sqrt {3}-15 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+468 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+4803 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+69648 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}-1813752}\right \}\)
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 \[ \left [\begin {array}{c} v_{1} \\ v_{2} \\ t \end {array}\right ] = t \left [\begin {array}{c} \frac {9 \sqrt {23}\, \left (3 i \sqrt {28985}\, \sqrt {3}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+92 i \sqrt {3}\, \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+2845 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}-4608 i \sqrt {3}\, \sqrt {28985}-41400 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \sqrt {3}-9 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+276 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+2845 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+41400 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}-1119744\right )}{23 \left (\sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}\right )-3 \cos \left (\frac {\arctan \left (\frac {81 \sqrt {86955}}{4651}\right )}{6}\right )\right ) \left (5 i \sqrt {28985}\, \sqrt {3}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+156 i \sqrt {3}\, \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+4803 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}-7464 i \sqrt {3}\, \sqrt {28985}-69648 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \sqrt {3}-15 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+468 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+4803 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+69648 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}-1813752\right )} \\ \frac {i \sqrt {28985}\, \sqrt {3}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+36 i \sqrt {3}\, \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+1071 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}-1104 i \sqrt {3}\, \sqrt {28985}-15096 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \sqrt {3}-3 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+108 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+1071 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+15096 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}-268272}{5 i \sqrt {28985}\, \sqrt {3}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+156 i \sqrt {3}\, \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+4803 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}} \sqrt {3}-7464 i \sqrt {3}\, \sqrt {28985}-69648 i \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}} \sqrt {3}-15 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+468 \sqrt {28985}\, \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+4803 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+69648 \left (2916+12 i \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}-1813752} \\ 1 \end {array}\right ] \] 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 |
| \(\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1\) | \(1\) | \(1\) | No | \(\left [\begin {array}{c} \frac {3 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}+\frac {828}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+81}{\left (\frac {7 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {322}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+21+\left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1\right )^{2}\right ) \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )} \\ \frac {2 \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1\right )^{2}-\frac {2 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{3}-\frac {184}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}-39}{\frac {7 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {322}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+21+\left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1\right )^{2}} \\ 1 \end {array}\right ]\) |
| \(-\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\) | \(1\) | \(1\) | No | \(\left [\begin {array}{c} \frac {-\frac {3 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{2}-\frac {414}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+81+9 i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{\left (-\frac {7 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {161}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+21+\frac {7 i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {2 \left (-\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{3}+\frac {92}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}-39-2 i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{-\frac {7 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {161}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+21+\frac {7 i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\) |
| \(-\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\) | \(1\) | \(1\) | No | \(\left [\begin {array}{c} \frac {-\frac {3 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{2}-\frac {414}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+81-9 i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{\left (-\frac {7 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {161}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+21-\frac {7 i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {2 \left (-\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{3}+\frac {92}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}-39+2 i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{-\frac {7 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {161}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+21-\frac {7 i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\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 homogeneous 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 \left (t \right ) \\ z \left (t \right ) \end {array}\right ] &= c_{1} \left [\begin {array}{c} \frac {9 \,{\mathrm e}^{\left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1\right ) t} \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{3}+\frac {92}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+9\right )}{\left (\frac {7 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {322}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+21+\left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1\right )^{2}\right ) \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )} \\ \frac {{\mathrm e}^{\left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1\right ) t} \left (2 \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1\right )^{2}-\frac {2 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{3}-\frac {184}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}-39\right )}{\frac {7 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {322}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+21+\left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1\right )^{2}} \\ {\mathrm e}^{\left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1\right ) t} \end {array}\right ] + c_{2} \text {Expression too large to display} + c_{3} \text {Expression too large to display} \end {align*}
Now that we found homogeneous solution above, we need to find a particular solution \(\vec {x}_p(t)\). We will use Variation of parameters. The fundamental matrix is \[ \Phi =\begin {bmatrix} \vec {x}_{1} & \vec {x}_{2} & \cdots \end {bmatrix} \] Where \(\vec {x}_i\) are the solution basis found above. Therefore the fundamental matrix is \begin {align*} \Phi (t)&= \text {Expression too large to display} \end {align*}
The particular solution is then given by \begin {align*} \vec {x}_p(t) &= \Phi \int { \Phi ^{-1} \vec {G}(t) \, dt}\\ \end {align*}
But \begin {align*} \Phi ^{-1} &= \text {Expression too large to display} \end {align*}
Hence \begin {align*} \vec {x}_p(t) &= \text {Expression too large to display} \int { \text {Expression too large to display} \left [\begin {array}{c} {\mathrm e}^{-t}-3 t \\ 2 \,{\mathrm e}^{-t}+t \\ 2 \,{\mathrm e}^{-t}-t \end {array}\right ] \, dt}\\ &= \text {Expression too large to display} \int { \text {Expression too large to display} \, dt}\\ &= \text {Expression too large to display} \text {Expression too large to display} \\ &= \text {Expression too large to display} \end {align*}
Now that we found particular solution, the final solution is \begin {align*} \vec {x}(t) &= \vec {x}_h(t) + \vec {x}_p(t)\\ \left [\begin {array}{c} x \left (t \right ) \\ y \left (t \right ) \\ z \left (t \right ) \end {array}\right ] &= \left [\begin {array}{c} \frac {9 c_{1} {\mathrm e}^{\left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1\right ) t} \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{3}+\frac {92}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+9\right )}{\left (\frac {7 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {322}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+21+\left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1\right )^{2}\right ) \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}\right )} \\ \frac {c_{1} {\mathrm e}^{\left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1\right ) t} \left (2 \left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1\right )^{2}-\frac {2 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{3}-\frac {184}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}-39\right )}{\frac {7 \left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {322}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+21+\left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1\right )^{2}} \\ c_{1} {\mathrm e}^{\left (\frac {\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 i \sqrt {86955}\right )^{\frac {1}{3}}}+1\right ) t} \end {array}\right ] + \text {Expression too large to display} + \text {Expression too large to display} + \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 )=x \left (t \right )-y \left (t \right )+2 z \left (t \right )+\frac {1}{{\mathrm e}^{t}}-3 t , y^{\prime }\left (t \right )=3 x \left (t \right )-4 y \left (t \right )+z \left (t \right )+\frac {2}{{\mathrm e}^{t}}+t , z^{\prime }\left (t \right )=-2 x \left (t \right )+5 y \left (t \right )+6 z \left (t \right )+\frac {2}{{\mathrm e}^{t}}-t \right ] \\ \bullet & {} & \textrm {Define vector}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}\left (t \right )=\left [\begin {array}{c} x \left (t \right ) \\ y \left (t \right ) \\ 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} 1 & -1 & 2 \\ 3 & -4 & 1 \\ -2 & 5 & 6 \end {array}\right ]\cdot {\moverset {\rightarrow }{x}}\left (t \right )+\left [\begin {array}{c} \frac {x \left (t \right ) {\mathrm e}^{t}-y \left (t \right ) {\mathrm e}^{t}+2 z \left (t \right ) {\mathrm e}^{t}-3 \,{\mathrm e}^{t} t +1}{{\mathrm e}^{t}}-x \left (t \right )+y \left (t \right )-2 z \left (t \right ) \\ \frac {3 x \left (t \right ) {\mathrm e}^{t}-4 y \left (t \right ) {\mathrm e}^{t}+z \left (t \right ) {\mathrm e}^{t}+{\mathrm e}^{t} t +2}{{\mathrm e}^{t}}-3 x \left (t \right )+4 y \left (t \right )-z \left (t \right ) \\ -\frac {2 x \left (t \right ) {\mathrm e}^{t}-5 y \left (t \right ) {\mathrm e}^{t}-6 z \left (t \right ) {\mathrm e}^{t}+{\mathrm e}^{t} t -2}{{\mathrm e}^{t}}+2 x \left (t \right )-5 y \left (t \right )-6 z \left (t \right ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}^{\prime }\left (t \right )=\left [\begin {array}{ccc} 1 & -1 & 2 \\ 3 & -4 & 1 \\ -2 & 5 & 6 \end {array}\right ]\cdot {\moverset {\rightarrow }{x}}\left (t \right ) \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{ccc} 1 & -1 & 2 \\ 3 & -4 & 1 \\ -2 & 5 & 6 \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 (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1, \left [\begin {array}{c} \frac {9 \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{3}+\frac {92}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+9\right )}{\left (\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {322}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21+\left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1\right )^{2}\right ) \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )} \\ \frac {2 \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1\right )^{2}-\frac {2 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{3}-\frac {184}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}-39}{\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {322}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21+\left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1\right )^{2}} \\ 1 \end {array}\right ]\right ], \left [-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {9 \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+9-\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )\right )}{\left (-\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {161}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21-\frac {7 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {2 \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{3}+\frac {92}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}-39+2 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{-\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {161}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21-\frac {7 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\right ], \left [-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {9 \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+9+\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )\right )}{\left (-\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {161}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21+\frac {7 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {2 \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{3}+\frac {92}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}-39-2 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{-\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {161}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21+\frac {7 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1, \left [\begin {array}{c} \frac {9 \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{3}+\frac {92}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+9\right )}{\left (\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {322}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21+\left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1\right )^{2}\right ) \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )} \\ \frac {2 \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1\right )^{2}-\frac {2 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{3}-\frac {184}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}-39}{\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {322}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21+\left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1\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 (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1\right ) t}\cdot \left [\begin {array}{c} \frac {9 \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{3}+\frac {92}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+9\right )}{\left (\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {322}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21+\left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1\right )^{2}\right ) \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )} \\ \frac {2 \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1\right )^{2}-\frac {2 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{3}-\frac {184}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}-39}{\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {322}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21+\left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1\right )^{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {9 \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+9-\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )\right )}{\left (-\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {161}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21-\frac {7 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {2 \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{3}+\frac {92}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}-39+2 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{-\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {161}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21-\frac {7 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\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 (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}\cdot \\ \left [\begin {array}{c} \frac {9 \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+9-\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )\right )}{\left (-\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {161}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21-\frac {7 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {2 \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{3}+\frac {92}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}-39+2 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{-\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {161}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21-\frac {7 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {9 \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+9+\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )\right )}{\left (-\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {161}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21+\frac {7 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {2 \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{3}+\frac {92}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}-39-2 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{-\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {161}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21+\frac {7 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\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 (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}\cdot \\ \left [\begin {array}{c} \frac {9 \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+9+\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )\right )}{\left (-\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {161}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21+\frac {7 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {2 \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{3}+\frac {92}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}-39-2 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{-\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {161}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21+\frac {7 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\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 (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1\right ) t}\cdot \left [\begin {array}{c} \frac {9 \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{3}+\frac {92}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+9\right )}{\left (\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {322}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21+\left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1\right )^{2}\right ) \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )} \\ \frac {2 \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1\right )^{2}-\frac {2 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{3}-\frac {184}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}-39}{\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {322}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21+\left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}+\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1\right )^{2}} \\ 1 \end {array}\right ]+ \\ c_{2} {\mathrm e}^{\left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}\cdot \\ \left [\begin {array}{c} \frac {9 \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+9-\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )\right )}{\left (-\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {161}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21-\frac {7 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {2 \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{3}+\frac {92}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}-39+2 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{-\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {161}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21-\frac {7 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ] \\ + \\ c_{3} {\mathrm e}^{\left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}\cdot \\ \left [\begin {array}{c} \frac {9 \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+9+\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )\right )}{\left (-\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {161}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21+\frac {7 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {2 \left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{3}+\frac {92}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}-39-2 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{-\frac {7 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {161}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+21+\frac {7 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{12}-\frac {23}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}{6}-\frac {46}{\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}}\right )}{2}\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 \left (t \right ) \\ z \left (t \right ) \end {array}\right ]=\left [\begin {array}{c} -\frac {302315 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {2}{3}} \left (c_{1} {\mathrm e}^{\frac {\left (\left (2916+12 \,\mathrm {I} \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+6 \left (2916+12 \,\mathrm {I} \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+276\right ) t}{6 \left (2916+12 \,\mathrm {I} \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}}} \left (-\frac {1522464 \,\mathrm {I}}{23255}+\frac {\left (223547 \,\mathrm {I}-\frac {2159 \sqrt {86955}}{3}\right ) \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {2}{3}}}{1209260}+\left (\mathrm {I}+\frac {81 \sqrt {86955}}{4651}\right ) \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}-\frac {26402 \sqrt {86955}}{302315}\right )+c_{2} {\mathrm e}^{-\frac {\left (\sqrt {3}\, \sqrt {23}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )-3 \sqrt {23}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )-3\right ) t}{3}} \left (\frac {761232 \,\mathrm {I}}{23255}+\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {2}{3}} \left (-223547 \,\mathrm {I}+223547 \sqrt {3}+\frac {2159 \sqrt {86955}}{3}+2159 \,\mathrm {I} \sqrt {28985}\right )}{2418520}+\left (\mathrm {I}+\frac {81 \sqrt {86955}}{4651}\right ) \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}-\frac {39603 \,\mathrm {I} \sqrt {28985}}{302315}+\frac {13201 \sqrt {86955}}{302315}+\frac {761232 \sqrt {3}}{23255}\right )+{\mathrm e}^{-\frac {\left (\sqrt {3}\, \sqrt {23}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )+3 \sqrt {23}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )-3\right ) t}{3}} c_{3} \left (\frac {761232 \,\mathrm {I}}{23255}+\frac {\left (-223547 \,\mathrm {I}-223547 \sqrt {3}+\frac {2159 \sqrt {86955}}{3}-2159 \,\mathrm {I} \sqrt {28985}\right ) \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {2}{3}}}{2418520}+\left (\mathrm {I}+\frac {81 \sqrt {86955}}{4651}\right ) \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}+\frac {39603 \,\mathrm {I} \sqrt {28985}}{302315}+\frac {13201 \sqrt {86955}}{302315}-\frac {761232 \sqrt {3}}{23255}\right )\right )}{2088 \left (3758 \sqrt {86955}+2043387 \,\mathrm {I}\right )} \\ \frac {-749316 c_{1} \left (\left (\mathrm {I}-\frac {2273 \sqrt {86955}}{187329}\right ) \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}+\left (\frac {18643 \,\mathrm {I}}{374658}-\frac {277 \sqrt {86955}}{374658}\right ) \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {2}{3}}+\frac {1292978 \,\mathrm {I}}{62443}+\frac {22518 \sqrt {86955}}{62443}\right ) {\mathrm e}^{\frac {\left (\left (2916+12 \,\mathrm {I} \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+6 \left (2916+12 \,\mathrm {I} \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+276\right ) t}{6 \left (2916+12 \,\mathrm {I} \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}}}-13638 \left (\left (\mathrm {I} \sqrt {28985}-\frac {62443 \,\mathrm {I}}{2273}+\frac {62443 \sqrt {3}}{2273}+\frac {\sqrt {86955}}{3}\right ) \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}+\left (-\frac {277 \,\mathrm {I} \sqrt {28985}}{4546}-\frac {18643 \,\mathrm {I}}{13638}-\frac {18643 \sqrt {3}}{13638}+\frac {277 \sqrt {86955}}{13638}\right ) \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {2}{3}}+\frac {2585956 \,\mathrm {I}}{2273}+\frac {45036 \sqrt {86955}}{2273}\right ) c_{2} {\mathrm e}^{-\frac {\left (\sqrt {3}\, \sqrt {23}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )-3 \sqrt {23}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )-3\right ) t}{3}}+13638 \,{\mathrm e}^{-\frac {\left (\sqrt {3}\, \sqrt {23}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )+3 \sqrt {23}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )-3\right ) t}{3}} c_{3} \left (\left (\mathrm {I} \sqrt {28985}+\frac {62443 \,\mathrm {I}}{2273}+\frac {62443 \sqrt {3}}{2273}-\frac {\sqrt {86955}}{3}\right ) \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}+\left (-\frac {277 \,\mathrm {I} \sqrt {28985}}{4546}+\frac {18643 \,\mathrm {I}}{13638}-\frac {18643 \sqrt {3}}{13638}-\frac {277 \sqrt {86955}}{13638}\right ) \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {2}{3}}-\frac {2585956 \,\mathrm {I}}{2273}-\frac {45036 \sqrt {86955}}{2273}\right )}{169128 \sqrt {86955}+9711288 \,\mathrm {I}} \\ {\mathrm e}^{\frac {\left (\left (2916+12 \,\mathrm {I} \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+6 \left (2916+12 \,\mathrm {I} \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+276\right ) t}{6 \left (2916+12 \,\mathrm {I} \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}}} c_{1} +{\mathrm e}^{-\frac {\left (\sqrt {3}\, \sqrt {23}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )-3 \sqrt {23}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )-3\right ) t}{3}} c_{2} +{\mathrm e}^{-\frac {\left (\sqrt {3}\, \sqrt {23}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )+3 \sqrt {23}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )-3\right ) t}{3}} c_{3} \end {array}\right ] \\ \bullet & {} & \textrm {Solution to the system of ODEs}\hspace {3pt} \\ {} & {} & \left \{x \left (t \right )=-\frac {302315 \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {2}{3}} \left (c_{1} {\mathrm e}^{\frac {\left (\left (2916+12 \,\mathrm {I} \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+6 \left (2916+12 \,\mathrm {I} \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+276\right ) t}{6 \left (2916+12 \,\mathrm {I} \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}}} \left (-\frac {1522464 \,\mathrm {I}}{23255}+\frac {\left (223547 \,\mathrm {I}-\frac {2159 \sqrt {86955}}{3}\right ) \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {2}{3}}}{1209260}+\left (\mathrm {I}+\frac {81 \sqrt {86955}}{4651}\right ) \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}-\frac {26402 \sqrt {86955}}{302315}\right )+c_{2} {\mathrm e}^{-\frac {\left (\sqrt {3}\, \sqrt {23}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )-3 \sqrt {23}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )-3\right ) t}{3}} \left (\frac {761232 \,\mathrm {I}}{23255}+\frac {\left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {2}{3}} \left (-223547 \,\mathrm {I}+223547 \sqrt {3}+\frac {2159 \sqrt {86955}}{3}+2159 \,\mathrm {I} \sqrt {28985}\right )}{2418520}+\left (\mathrm {I}+\frac {81 \sqrt {86955}}{4651}\right ) \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}-\frac {39603 \,\mathrm {I} \sqrt {28985}}{302315}+\frac {13201 \sqrt {86955}}{302315}+\frac {761232 \sqrt {3}}{23255}\right )+{\mathrm e}^{-\frac {\left (\sqrt {3}\, \sqrt {23}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )+3 \sqrt {23}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )-3\right ) t}{3}} c_{3} \left (\frac {761232 \,\mathrm {I}}{23255}+\frac {\left (-223547 \,\mathrm {I}-223547 \sqrt {3}+\frac {2159 \sqrt {86955}}{3}-2159 \,\mathrm {I} \sqrt {28985}\right ) \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {2}{3}}}{2418520}+\left (\mathrm {I}+\frac {81 \sqrt {86955}}{4651}\right ) \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}+\frac {39603 \,\mathrm {I} \sqrt {28985}}{302315}+\frac {13201 \sqrt {86955}}{302315}-\frac {761232 \sqrt {3}}{23255}\right )\right )}{2088 \left (3758 \sqrt {86955}+2043387 \,\mathrm {I}\right )}, y \left (t \right )=\frac {-749316 c_{1} \left (\left (\mathrm {I}-\frac {2273 \sqrt {86955}}{187329}\right ) \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}+\left (\frac {18643 \,\mathrm {I}}{374658}-\frac {277 \sqrt {86955}}{374658}\right ) \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {2}{3}}+\frac {1292978 \,\mathrm {I}}{62443}+\frac {22518 \sqrt {86955}}{62443}\right ) {\mathrm e}^{\frac {\left (\left (2916+12 \,\mathrm {I} \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+6 \left (2916+12 \,\mathrm {I} \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+276\right ) t}{6 \left (2916+12 \,\mathrm {I} \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}}}-13638 \left (\left (\mathrm {I} \sqrt {28985}-\frac {62443 \,\mathrm {I}}{2273}+\frac {62443 \sqrt {3}}{2273}+\frac {\sqrt {86955}}{3}\right ) \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}+\left (-\frac {277 \,\mathrm {I} \sqrt {28985}}{4546}-\frac {18643 \,\mathrm {I}}{13638}-\frac {18643 \sqrt {3}}{13638}+\frac {277 \sqrt {86955}}{13638}\right ) \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {2}{3}}+\frac {2585956 \,\mathrm {I}}{2273}+\frac {45036 \sqrt {86955}}{2273}\right ) c_{2} {\mathrm e}^{-\frac {\left (\sqrt {3}\, \sqrt {23}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )-3 \sqrt {23}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )-3\right ) t}{3}}+13638 \,{\mathrm e}^{-\frac {\left (\sqrt {3}\, \sqrt {23}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )+3 \sqrt {23}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )-3\right ) t}{3}} c_{3} \left (\left (\mathrm {I} \sqrt {28985}+\frac {62443 \,\mathrm {I}}{2273}+\frac {62443 \sqrt {3}}{2273}-\frac {\sqrt {86955}}{3}\right ) \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {1}{3}}+\left (-\frac {277 \,\mathrm {I} \sqrt {28985}}{4546}+\frac {18643 \,\mathrm {I}}{13638}-\frac {18643 \sqrt {3}}{13638}-\frac {277 \sqrt {86955}}{13638}\right ) \left (2916+12 \,\mathrm {I} \sqrt {86955}\right )^{\frac {2}{3}}-\frac {2585956 \,\mathrm {I}}{2273}-\frac {45036 \sqrt {86955}}{2273}\right )}{169128 \sqrt {86955}+9711288 \,\mathrm {I}}, z \left (t \right )={\mathrm e}^{\frac {\left (\left (2916+12 \,\mathrm {I} \sqrt {3}\, \sqrt {28985}\right )^{\frac {2}{3}}+6 \left (2916+12 \,\mathrm {I} \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}+276\right ) t}{6 \left (2916+12 \,\mathrm {I} \sqrt {3}\, \sqrt {28985}\right )^{\frac {1}{3}}}} c_{1} +{\mathrm e}^{-\frac {\left (\sqrt {3}\, \sqrt {23}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )-3 \sqrt {23}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )-3\right ) t}{3}} c_{2} +{\mathrm e}^{-\frac {\left (\sqrt {3}\, \sqrt {23}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )+3 \sqrt {23}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {86955}}{243}\right )}{3}\right )-3\right ) t}{3}} c_{3} \right \} \end {array} \]
✗ Solution by Maple
dsolve([diff(x(t),t)=x(t)-y(t)+2*z(t)+exp(-t)-3*t,diff(y(t),t)=3*x(t)-4*y(t)+z(t)+2*exp(-t)+t,diff(z(t),t)=-2*x(t)+5*y(t)+6*z(t)+2*exp(-t)-t],singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.191 (sec). Leaf size: 3251
DSolve[{x'[t]==x[t]-y[t]+2*z[t]+Exp[-t]-3*t,y'[t]==3*x[t]-4*y[t]+z[t]+2*Exp[-t]+t,z'[t]==-2*x[t]+5*y[t]+6*z[t]+2*Exp[-t]-t},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
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