problem 1908

Internal problem ID [10230]
Internal ﬁle name [OUTPUT/9177_Monday_June_06_2022_01_34_22_PM_80984372/index.tex]

Book: Diﬀerential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of ﬁrst order odes
Problem number: 1908.
ODE order: 1.
ODE degree: 1.

Solve \begin {align*} x^{\prime }\left (t \right )&=6 x \left (t \right )-72 y \left (t \right )+44 z \left (t \right )\\ y^{\prime }\left (t \right )&=4 x \left (t \right )-4 y \left (t \right )+26 z \left (t \right )\\ z^{\prime }\left (t \right )&=6 x \left (t \right )-63 y \left (t \right )+38 z \left (t \right ) \end {align*}

9.53.1 Solution using Matrix exponential method

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 \begin {align*} \vec {x}'(t) &= A\, \vec {x}(t) \end {align*}

Or \begin {align*} \left [\begin {array}{c} x^{\prime }\left (t \right ) \\ y^{\prime }\left (t \right ) \\ z^{\prime }\left (t \right ) \end {array}\right ] &= \left [\begin {array}{ccc} 6 & -72 & 44 \\ 4 & -4 & 26 \\ 6 & -63 & 38 \end {array}\right ]\, \left [\begin {array}{c} x \left (t \right ) \\ y \left (t \right ) \\ z \left (t \right ) \end {array}\right ] \end {align*}

For the above matrix $A$, the matrix exponential can be found to be \begin {align*} e^{A t} &= \text {Expression too large to display}\\ &= \text {Expression too large to display} \end {align*}

Therefore the homogeneous solution is \begin {align*} \vec {x}_h(t) &= e^{A t} \vec {c} \\ &= \text {Expression too large to display} \left [\begin {array}{c} c_{1} \\ c_{2} \\ c_{3} \end {array}\right ] \\ &= \text {Expression too large to display}\\ &= \text {Expression too large to display} \end {align*}

Since no forcing function is given, then the ﬁnal solution is $\vec {x}_h(t)$ above.

9.53.2 Solution using explicit Eigenvalue and Eigenvector method

This is a system of linear ODE’s given as \begin {align*} \vec {x}'(t) &= A\, \vec {x}(t) \end {align*}

The ﬁrst step is ﬁnd the homogeneous solution. We start by ﬁnding the eigenvalues of $A$. This is done by solving the following equation for the eigenvalues $\lambda $ \begin {align*} \operatorname {det} \left ( A- \lambda I \right ) &= 0 \end {align*}

Expanding gives \begin {align*} \operatorname {det} \left (\left [\begin {array}{ccc} 6 & -72 & 44 \\ 4 & -4 & 26 \\ 6 & -63 & 38 \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} 6-\lambda & -72 & 44 \\ 4 & -4-\lambda & 26 \\ 6 & -63 & 38-\lambda \end {array}\right ]\right ) &= 0 \end {align*}

Which gives the characteristic equation \begin {align*} \lambda ^{3}-40 \lambda ^{2}+1714 \lambda +1404&=0 \end {align*}

The roots of the above are the eigenvalues. \begin {align*} \lambda _1 &= -\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}\\ \lambda _2 &= \frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\\ \lambda _3 &= \frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}


eigenvalue	algebraic multiplicity	type of eigenvalue

$-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}$	$1$	real eigenvalue

$\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}$	$1$	complex eigenvalue

$\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}$	$1$	complex eigenvalue

Considering the eigenvalue $\lambda _{1} = -\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}$

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} 6 & -72 & 44 \\ 4 & -4 & 26 \\ 6 & -63 & 38 \end {array}\right ] - \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}\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 (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}} & -72 & 44 \\ 4 & \frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-52 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}} & 26 \\ 6 & -63 & \frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+74 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542}{3 \left (263474+18 \sqrt {351406311}\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*}

\begin {align*} R_{3} = R_{3}-\frac {18 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}} R_{1}}{\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}&-72&44&0\\ 0&-\frac {6 \left (\left (\sqrt {351406311}+29199\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-74 \sqrt {351406311}-186 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-386184\right )}{\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}} \left (-\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+3542\right )}&\frac {26 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-1100 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-92092}{\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542}&0\\ 0&\frac {-63 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+2682 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+223146}{\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542}&-\frac {6 \left (\left (\sqrt {351406311}+4405\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+52 \sqrt {351406311}-616 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+1458134\right )}{\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}} \left (-\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+3542\right )}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}+\frac {\left (-63 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+2682 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+223146\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}} \left (-\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+3542\right ) R_{2}}{6 \left (\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542\right ) \left (\left (\sqrt {351406311}+29199\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-74 \sqrt {351406311}-186 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-386184\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}&-72&44&0\\ 0&-\frac {6 \left (\left (\sqrt {351406311}+29199\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-74 \sqrt {351406311}-186 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-386184\right )}{\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}} \left (-\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+3542\right )}&\frac {26 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-1100 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-92092}{\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542}&0\\ 0&0&0&0 \end {array} \right ] \end {align*}

Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} \frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}} & -72 & 44 \\ 0 & -\frac {6 \left (\left (\sqrt {351406311}+29199\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-74 \sqrt {351406311}-186 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-386184\right )}{\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}} \left (-\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+3542\right )} & \frac {26 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-1100 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-92092}{\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542} \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] 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 \(\left \{v_{1} = -\frac {12 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}} t \left (5346 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-11 \sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-44913 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-590 \sqrt {351406311}-16302948\right )}{\left (186 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-\sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-29199 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+74 \sqrt {351406311}+386184\right ) \left (\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542\right )}, v_{2} = -\frac {2 t \left (275 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+23023 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-117 \sqrt {351406311}-1712581\right )}{3 \left (186 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-\sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-29199 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+74 \sqrt {351406311}+386184\right )}\right \}\)

Hence the solution is \[ \left [\begin {array}{c} -\frac {12 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}} t \left (5346 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-11 \sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-44913 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-590 \sqrt {351406311}-16302948\right )}{\left (186 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-\sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-29199 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+74 \sqrt {351406311}+386184\right ) \left (\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542\right )} \\ -\frac {2 t \left (275 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+23023 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-117 \sqrt {351406311}-1712581\right )}{3 \left (186 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-\sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-29199 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+74 \sqrt {351406311}+386184\right )} \\ t \end {array}\right ] = \left [\begin {array}{c} -\frac {12 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}} t \left (5346 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-11 \sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-44913 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-590 \sqrt {351406311}-16302948\right )}{\left (186 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-\sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-29199 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+74 \sqrt {351406311}+386184\right ) \left (\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542\right )} \\ -\frac {2 t \left (275 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+23023 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-117 \sqrt {351406311}-1712581\right )}{3 \left (186 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-\sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-29199 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+74 \sqrt {351406311}+386184\right )} \\ t \end {array}\right ] \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \left [\begin {array}{c} -\frac {12 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}} t \left (5346 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-11 \sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-44913 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-590 \sqrt {351406311}-16302948\right )}{\left (186 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-\sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-29199 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+74 \sqrt {351406311}+386184\right ) \left (\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542\right )} \\ -\frac {2 t \left (275 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+23023 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-117 \sqrt {351406311}-1712581\right )}{3 \left (186 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-\sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-29199 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+74 \sqrt {351406311}+386184\right )} \\ t \end {array}\right ] = t \left [\begin {array}{c} -\frac {12 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}} \left (5346 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-11 \sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-44913 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-590 \sqrt {351406311}-16302948\right )}{\left (186 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-\sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-29199 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+74 \sqrt {351406311}+386184\right ) \left (\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542\right )} \\ -\frac {2 \left (275 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+23023 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-117 \sqrt {351406311}-1712581\right )}{3 \left (186 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-\sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-29199 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+74 \sqrt {351406311}+386184\right )} \\ 1 \end {array}\right ] \] Let $t = 1$ the eigenvector becomes \[ \left [\begin {array}{c} -\frac {12 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}} t \left (5346 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-11 \sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-44913 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-590 \sqrt {351406311}-16302948\right )}{\left (186 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-\sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-29199 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+74 \sqrt {351406311}+386184\right ) \left (\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542\right )} \\ -\frac {2 t \left (275 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+23023 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-117 \sqrt {351406311}-1712581\right )}{3 \left (186 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-\sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-29199 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+74 \sqrt {351406311}+386184\right )} \\ t \end {array}\right ] = \left [\begin {array}{c} -\frac {12 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}} \left (5346 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-11 \sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-44913 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-590 \sqrt {351406311}-16302948\right )}{\left (186 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-\sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-29199 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+74 \sqrt {351406311}+386184\right ) \left (\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542\right )} \\ -\frac {2 \left (275 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+23023 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-117 \sqrt {351406311}-1712581\right )}{3 \left (186 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-\sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-29199 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+74 \sqrt {351406311}+386184\right )} \\ 1 \end {array}\right ] \] Which is normalized to \[ \left [\begin {array}{c} -\frac {12 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}} t \left (5346 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-11 \sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-44913 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-590 \sqrt {351406311}-16302948\right )}{\left (186 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-\sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-29199 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+74 \sqrt {351406311}+386184\right ) \left (\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542\right )} \\ -\frac {2 t \left (275 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+23023 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-117 \sqrt {351406311}-1712581\right )}{3 \left (186 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-\sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-29199 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+74 \sqrt {351406311}+386184\right )} \\ t \end {array}\right ] = \left [\begin {array}{c} -\frac {12 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}} \left (5346 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-11 \sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-44913 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-590 \sqrt {351406311}-16302948\right )}{\left (186 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-\sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-29199 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+74 \sqrt {351406311}+386184\right ) \left (\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-22 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542\right )} \\ -\frac {2 \left (275 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+23023 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-117 \sqrt {351406311}-1712581\right )}{3 \left (186 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-\sqrt {351406311}\, \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-29199 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+74 \sqrt {351406311}+386184\right )} \\ 1 \end {array}\right ] \] Considering the eigenvalue $\lambda _{2} = \frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}$

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} 6 & -72 & 44 \\ 4 & -4 & 26 \\ 6 & -63 & 38 \end {array}\right ] - \left (\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\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 {\left (1+i \sqrt {3}\right ) \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542 i \sqrt {3}+44 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}-3542}{6 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}} & -72 & 44 \\ 4 & -\frac {\left (1+i \sqrt {3}\right ) \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542 i \sqrt {3}+104 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}-3542}{6 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}} & 26 \\ 6 & -63 & -\frac {\left (1+i \sqrt {3}\right ) \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542 i \sqrt {3}-148 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}-3542}{6 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\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 \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {22}{3}-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}+\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}&-72&44&0\\ 4&-\frac {52}{3}-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}+\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}&26&0\\ 6&-63&\frac {74}{3}-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}+\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}-\frac {4 R_{1}}{-\frac {22}{3}-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}+\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\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 {\left (1+i \sqrt {3}\right ) \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542 i \sqrt {3}+44 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}-3542}{6 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}}&-72&44&0\\ 0&\frac {6 \left (29199-29199 i \sqrt {3}-3 i \sqrt {117135437}+\sqrt {351406311}\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-2317104-2317104 i \sqrt {3}-1332 i \sqrt {117135437}-444 \sqrt {351406311}+2232 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}}{\left (44 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+\left (1+i \sqrt {3}\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+3542 i \sqrt {3}-3542\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}&\frac {-92092+2200 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}+26 i \left (\left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542\right ) \sqrt {3}+26 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}}{-3542+44 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}+i \left (\left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542\right ) \sqrt {3}+\left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}}&0\\ 6&-63&-\frac {\left (1+i \sqrt {3}\right ) \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542 i \sqrt {3}-148 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}-3542}{6 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}+\frac {36 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}} R_{1}}{\left (1+i \sqrt {3}\right ) \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542 i \sqrt {3}+44 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}-3542} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {\left (1+i \sqrt {3}\right ) \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542 i \sqrt {3}+44 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}-3542}{6 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}}&-72&44&0\\ 0&\frac {6 \left (29199-29199 i \sqrt {3}-3 i \sqrt {117135437}+\sqrt {351406311}\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-2317104-2317104 i \sqrt {3}-1332 i \sqrt {117135437}-444 \sqrt {351406311}+2232 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}}{\left (44 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+\left (1+i \sqrt {3}\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+3542 i \sqrt {3}-3542\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}&\frac {-92092+2200 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}+26 i \left (\left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542\right ) \sqrt {3}+26 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}}{-3542+44 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}+i \left (\left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542\right ) \sqrt {3}+\left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}}&0\\ 0&-\frac {9 \left (7 i \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}} \sqrt {3}+7 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+24794 i \sqrt {3}+596 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-24794\right )}{i \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}} \sqrt {3}+\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+3542 i \sqrt {3}+44 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542}&\frac {6 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {117135437}-26430 i \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}} \sqrt {3}-18 i \sqrt {117135437}\, \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}+7392 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+312 \sqrt {3}\, \sqrt {117135437}+8748804 i \sqrt {3}+936 i \sqrt {117135437}+26430 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}+8748804}{\left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}} \left (i \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}} \sqrt {3}+\left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542 i \sqrt {3}+44 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}-3542\right )}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}+\frac {3 \left (7 i \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}} \sqrt {3}+7 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+24794 i \sqrt {3}+596 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-24794\right ) \left (44 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+\left (1+i \sqrt {3}\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+3542 i \sqrt {3}-3542\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}} R_{2}}{2 \left (i \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}} \sqrt {3}+\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+3542 i \sqrt {3}+44 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542\right ) \left (\left (29199-29199 i \sqrt {3}-3 i \sqrt {117135437}+\sqrt {351406311}\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-386184-386184 i \sqrt {3}-222 i \sqrt {117135437}-74 \sqrt {351406311}+372 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {\left (1+i \sqrt {3}\right ) \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542 i \sqrt {3}+44 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}-3542}{6 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}}&-72&44&0\\ 0&\frac {6 \left (29199-29199 i \sqrt {3}-3 i \sqrt {117135437}+\sqrt {351406311}\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-2317104-2317104 i \sqrt {3}-1332 i \sqrt {117135437}-444 \sqrt {351406311}+2232 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}}{\left (44 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+\left (1+i \sqrt {3}\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+3542 i \sqrt {3}-3542\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}&\frac {-92092+2200 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}+26 i \left (\left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542\right ) \sqrt {3}+26 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}}{-3542+44 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}+i \left (\left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542\right ) \sqrt {3}+\left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}}&0\\ 0&0&0&0 \end {array} \right ] \end {align*}

Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} -\frac {\left (1+i \sqrt {3}\right ) \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542 i \sqrt {3}+44 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}-3542}{6 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}} & -72 & 44 \\ 0 & \frac {6 \left (29199-29199 i \sqrt {3}-3 i \sqrt {117135437}+\sqrt {351406311}\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-2317104-2317104 i \sqrt {3}-1332 i \sqrt {117135437}-444 \sqrt {351406311}+2232 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}}{\left (44 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+\left (1+i \sqrt {3}\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+3542 i \sqrt {3}-3542\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}} & \frac {-92092+2200 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}+26 i \left (\left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542\right ) \sqrt {3}+26 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}}{-3542+44 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}+i \left (\left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542\right ) \sqrt {3}+\left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}} \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are $\{v_{3}\}$ and the leading variables are $\{v_{1}, v_{2}\}$. Let $v_{3} = t$. Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation $\text {Expression too large to display}$

Hence the solution is \[ \text {Expression too large to display} \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \text {Expression too large to display} \text {Expression too large to display} \] Let $t = 1$ the eigenvector becomes \[ \text {Expression too large to display} \] Which is normalized to \[ \text {Expression too large to display} \] Considering the eigenvalue $\lambda _{3} = \frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}$

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} 6 & -72 & 44 \\ 4 & -4 & 26 \\ 6 & -63 & 38 \end {array}\right ] - \left (\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\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 {\left (-1+i \sqrt {3}\right ) \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542 i \sqrt {3}-44 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}+3542}{6 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}} & -72 & 44 \\ 4 & \frac {\left (-1+i \sqrt {3}\right ) \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542 i \sqrt {3}-104 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}+3542}{6 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}} & 26 \\ 6 & -63 & \frac {\left (-1+i \sqrt {3}\right ) \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542 i \sqrt {3}+148 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}+3542}{6 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\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 \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {22}{3}-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}+\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}&-72&44&0\\ 4&-\frac {52}{3}-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}+\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}&26&0\\ 6&-63&\frac {74}{3}-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}+\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}-\frac {4 R_{1}}{-\frac {22}{3}-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}+\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\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 {\left (-1+i \sqrt {3}\right ) \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542 i \sqrt {3}-44 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}+3542}{6 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}}&-72&44&0\\ 0&\frac {-6 \left (29199 i \sqrt {3}+3 i \sqrt {117135437}+\sqrt {351406311}+29199\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+2317104-2317104 i \sqrt {3}-1332 i \sqrt {117135437}+444 \sqrt {351406311}-2232 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}}{\left (-44 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+\left (-1+i \sqrt {3}\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+3542 i \sqrt {3}+3542\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}&\frac {-92092 i+26 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )+92092 \sqrt {3}+2200 i \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}}{-3542 i+\left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )+3542 \sqrt {3}+44 i \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}}&0\\ 6&-63&\frac {\left (-1+i \sqrt {3}\right ) \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542 i \sqrt {3}+148 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}+3542}{6 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}-\frac {36 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}} R_{1}}{\left (-1+i \sqrt {3}\right ) \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542 i \sqrt {3}-44 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}+3542} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {\left (-1+i \sqrt {3}\right ) \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542 i \sqrt {3}-44 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}+3542}{6 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}}&-72&44&0\\ 0&\frac {-6 \left (29199 i \sqrt {3}+3 i \sqrt {117135437}+\sqrt {351406311}+29199\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+2317104-2317104 i \sqrt {3}-1332 i \sqrt {117135437}+444 \sqrt {351406311}-2232 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}}{\left (-44 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+\left (-1+i \sqrt {3}\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+3542 i \sqrt {3}+3542\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}&\frac {-92092 i+26 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )+92092 \sqrt {3}+2200 i \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}}{-3542 i+\left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )+3542 \sqrt {3}+44 i \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}}&0\\ 0&\frac {223146 i-63 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )-223146 \sqrt {3}-5364 i \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{-3542 i+\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )+3542 \sqrt {3}+44 i \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}&\frac {-6 \left (4405 i \sqrt {3}+3 i \sqrt {117135437}+\sqrt {351406311}+4405\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-8748804+8748804 i \sqrt {3}+936 i \sqrt {117135437}-312 \sqrt {351406311}-7392 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}}{\left (-44 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+\left (-1+i \sqrt {3}\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+3542 i \sqrt {3}+3542\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}-\frac {3 \left (24794 i-7 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )-24794 \sqrt {3}-596 i \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}\right ) \left (-44 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+\left (-1+i \sqrt {3}\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+3542 i \sqrt {3}+3542\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}} R_{2}}{2 \left (-3542 i+\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )+3542 \sqrt {3}+44 i \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}\right ) \left (-\left (29199 i \sqrt {3}+3 i \sqrt {117135437}+\sqrt {351406311}+29199\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+386184-386184 i \sqrt {3}-222 i \sqrt {117135437}+74 \sqrt {351406311}-372 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {\left (-1+i \sqrt {3}\right ) \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542 i \sqrt {3}-44 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}+3542}{6 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}}&-72&44&0\\ 0&\frac {-6 \left (29199 i \sqrt {3}+3 i \sqrt {117135437}+\sqrt {351406311}+29199\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+2317104-2317104 i \sqrt {3}-1332 i \sqrt {117135437}+444 \sqrt {351406311}-2232 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}}{\left (-44 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+\left (-1+i \sqrt {3}\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+3542 i \sqrt {3}+3542\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}&\frac {-92092 i+26 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )+92092 \sqrt {3}+2200 i \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}}{-3542 i+\left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )+3542 \sqrt {3}+44 i \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}}&0\\ 0&0&0&0 \end {array} \right ] \end {align*}

Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} \frac {\left (-1+i \sqrt {3}\right ) \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}}+3542 i \sqrt {3}-44 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}+3542}{6 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}} & -72 & 44 \\ 0 & \frac {-6 \left (29199 i \sqrt {3}+3 i \sqrt {117135437}+\sqrt {351406311}+29199\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+2317104-2317104 i \sqrt {3}-1332 i \sqrt {117135437}+444 \sqrt {351406311}-2232 \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}}{\left (-44 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+\left (-1+i \sqrt {3}\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+3542 i \sqrt {3}+3542\right ) \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}} & \frac {-92092 i+26 \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )+92092 \sqrt {3}+2200 i \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}}{-3542 i+\left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {2}{3}} \left (\sqrt {3}+i\right )+3542 \sqrt {3}+44 i \left (263474+18 \sqrt {3}\, \sqrt {117135437}\right )^{\frac {1}{3}}} \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are $\{v_{3}\}$ and the leading variables are $\{v_{1}, v_{2}\}$. Let $v_{3} = t$. Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation $\text {Expression too large to display}$

Hence the solution is \[ \text {Expression too large to display} \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \text {Expression too large to display} \text {Expression too large to display} \] Let $t = 1$ the eigenvector becomes \[ \text {Expression too large to display} \] Which is normalized to \[ \text {Expression too large to display} \] The following table gives a summary of this result. It shows for each eigenvalue the algebraic multiplicity $m$, and its geometric multiplicity $k$ and the eigenvectors associated with the eigenvalue. If $m>k$ then the eigenvalue is defective which means the number of normal linearly independent eigenvectors associated with this eigenvalue (called the geometric multiplicity $k$) does not equal the algebraic multiplicity $m$, and we need to determine an additional $m-k$ generalized eigenvectors for this eigenvalue.


	multiplicity

eigenvalue	algebraic $m$	geometric $k$	defective?	eigenvectors

$-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}$	$1$	$1$	No	\(\left [\begin {array}{c} -\frac {436 \left (-\frac {11 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {38962}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}-\frac {832}{3}\right )}{\left (3 \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}\right )^{2}+\frac {211 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {747362}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {3842}{3}\right ) \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {22}{3}\right )} \\ \frac {11 \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}\right )^{2}+\frac {374 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {1324708}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}-\frac {16142}{3}}{18 \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}\right )^{2}+422 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-\frac {1494724}{\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+7684} \\ 1 \end {array}\right ]\)

$\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}$	$1$	$1$	No	\(\left [\begin {array}{c} -\frac {436 \left (\frac {11 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {19481}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}-\frac {832}{3}+\frac {11 i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (3 \left (\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-\frac {211 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}+\frac {373681}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {3842}{3}-\frac {211 i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right ) \left (\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {22}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {11 \left (\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-\frac {187 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {662354}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}-\frac {16142}{3}-187 i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{18 \left (\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-211 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+\frac {747362}{\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+7684-633 i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )} \\ 1 \end {array}\right ]\)

$\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}$	$1$	$1$	No	\(\left [\begin {array}{c} -\frac {436 \left (\frac {11 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {19481}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}-\frac {832}{3}-\frac {11 i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (3 \left (\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-\frac {211 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}+\frac {373681}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {3842}{3}+\frac {211 i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right ) \left (\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {22}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {11 \left (\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-\frac {187 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {662354}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}-\frac {16142}{3}+187 i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{18 \left (\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-211 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+\frac {747362}{\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+7684+633 i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )} \\ 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. Since eigenvalue $-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}$ is real and distinct then the corresponding eigenvector solution is \begin {align*} \vec {x}_{1}(t) &= \vec {v}_{1} e^{\left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}\right ) t}\\ &= \left [\begin {array}{c} -\frac {436 \left (-\frac {11 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {38962}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}-\frac {832}{3}\right )}{\left (3 \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}\right )^{2}+\frac {211 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {747362}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {3842}{3}\right ) \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {22}{3}\right )} \\ \frac {11 \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}\right )^{2}+\frac {374 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {1324708}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}-\frac {16142}{3}}{18 \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}\right )^{2}+422 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-\frac {1494724}{\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+7684} \\ 1 \end {array}\right ] e^{\left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}\right ) t} \end {align*}

Therefore the ﬁnal solution is \begin {align*} \vec {x}_h(t) &= c_{1} \vec {x}_{1}(t) + c_{2} \vec {x}_{2}(t) + c_{3} \vec {x}_{3}(t) \end {align*}

Which is written as \begin {align*} \left [\begin {array}{c} x \left (t \right ) \\ y \left (t \right ) \\ z \left (t \right ) \end {array}\right ] &= c_{1} \left [\begin {array}{c} -\frac {436 \,{\mathrm e}^{\left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}\right ) t} \left (-\frac {11 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {38962}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}-\frac {832}{3}\right )}{\left (3 \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}\right )^{2}+\frac {211 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {747362}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {3842}{3}\right ) \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {22}{3}\right )} \\ \frac {{\mathrm e}^{\left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}\right ) t} \left (11 \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}\right )^{2}+\frac {374 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {1324708}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}-\frac {16142}{3}\right )}{18 \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}\right )^{2}+422 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-\frac {1494724}{\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+7684} \\ {\mathrm e}^{\left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}\right ) t} \end {array}\right ] + c_{2} \text {Expression too large to display} + c_{3} \text {Expression too large to display} \end {align*}

Which becomes \begin {align*} \text {Expression too large to display} \end {align*}

\begin{align*} x \left (t \right ) &= \cos \left (\frac {\left (\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+3542\right ) t \sqrt {3}\, 4^{\frac {1}{3}}}{12 \left (131737+9 \sqrt {351406311}\right )^{\frac {1}{3}}}\right ) {\mathrm e}^{\frac {\left (-3542+\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+80 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}\right ) t}{6 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}} c_{3} +\sin \left (\frac {\left (\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+3542\right ) t \sqrt {3}\, 4^{\frac {1}{3}}}{12 \left (131737+9 \sqrt {351406311}\right )^{\frac {1}{3}}}\right ) {\mathrm e}^{\frac {\left (-3542+\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}+80 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}\right ) t}{6 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}} c_{2} +c_{1} {\mathrm e}^{-\frac {\left (\left (263474+18 \sqrt {351406311}\right )^{\frac {2}{3}}-40 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-3542\right ) t}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}} \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align*}

\begin{align*} x(t)\to -36 c_2 \text {RootSum}\left [\text {$\#$1}^3-40 \text {$\#$1}^2+1714 \text {$\#$1}+1404\&,\frac {2 \text {$\#$1} e^{\text {$\#$1} t}+e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-80 \text {$\#$1}+1714}\&\right ]+4 c_3 \text {RootSum}\left [\text {$\#$1}^3-40 \text {$\#$1}^2+1714 \text {$\#$1}+1404\&,\frac {11 \text {$\#$1} e^{\text {$\#$1} t}-424 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-80 \text {$\#$1}+1714}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3-40 \text {$\#$1}^2+1714 \text {$\#$1}+1404\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-34 \text {$\#$1} e^{\text {$\#$1} t}+1486 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-80 \text {$\#$1}+1714}\&\right ] \\ y(t)\to 4 c_1 \text {RootSum}\left [\text {$\#$1}^3-40 \text {$\#$1}^2+1714 \text {$\#$1}+1404\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}+e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-80 \text {$\#$1}+1714}\&\right ]+2 c_3 \text {RootSum}\left [\text {$\#$1}^3-40 \text {$\#$1}^2+1714 \text {$\#$1}+1404\&,\frac {13 \text {$\#$1} e^{\text {$\#$1} t}+10 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-80 \text {$\#$1}+1714}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3-40 \text {$\#$1}^2+1714 \text {$\#$1}+1404\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-44 \text {$\#$1} e^{\text {$\#$1} t}-36 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-80 \text {$\#$1}+1714}\&\right ] \\ z(t)\to 6 c_1 \text {RootSum}\left [\text {$\#$1}^3-40 \text {$\#$1}^2+1714 \text {$\#$1}+1404\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-38 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-80 \text {$\#$1}+1714}\&\right ]-9 c_2 \text {RootSum}\left [\text {$\#$1}^3-40 \text {$\#$1}^2+1714 \text {$\#$1}+1404\&,\frac {7 \text {$\#$1} e^{\text {$\#$1} t}+6 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-80 \text {$\#$1}+1714}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3-40 \text {$\#$1}^2+1714 \text {$\#$1}+1404\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-2 \text {$\#$1} e^{\text {$\#$1} t}+264 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-80 \text {$\#$1}+1714}\&\right ] \\ \end{align*}


	multiplicity

eigenvalue	algebraic \(m\)	geometric \(k\)	defective?	eigenvectors

\(-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}\)	\(1\)	\(1\)	No	\(\left [\begin {array}{c} -\frac {436 \left (-\frac {11 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {38962}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}-\frac {832}{3}\right )}{\left (3 \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}\right )^{2}+\frac {211 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {747362}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {3842}{3}\right ) \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {22}{3}\right )} \\ \frac {11 \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}\right )^{2}+\frac {374 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {1324708}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}-\frac {16142}{3}}{18 \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}\right )^{2}+422 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}-\frac {1494724}{\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+7684} \\ 1 \end {array}\right ]\)

\(\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\)	\(1\)	\(1\)	No	\(\left [\begin {array}{c} -\frac {436 \left (\frac {11 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {19481}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}-\frac {832}{3}+\frac {11 i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (3 \left (\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-\frac {211 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}+\frac {373681}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {3842}{3}-\frac {211 i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right ) \left (\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {22}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {11 \left (\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-\frac {187 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {662354}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}-\frac {16142}{3}-187 i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{18 \left (\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-211 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+\frac {747362}{\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+7684-633 i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )} \\ 1 \end {array}\right ]\)

\(\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\)	\(1\)	\(1\)	No	\(\left [\begin {array}{c} -\frac {436 \left (\frac {11 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {19481}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}-\frac {832}{3}-\frac {11 i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (3 \left (\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-\frac {211 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}+\frac {373681}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {3842}{3}+\frac {211 i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right ) \left (\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {22}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {11 \left (\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-\frac {187 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}+\frac {662354}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}-\frac {16142}{3}+187 i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{18 \left (\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{6}-\frac {1771}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+\frac {40}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-211 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}+\frac {747362}{\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}+7684+633 i \sqrt {3}\, \left (-\frac {\left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}{3}-\frac {3542}{3 \left (263474+18 \sqrt {351406311}\right )^{\frac {1}{3}}}\right )} \\ 1 \end {array}\right ]\)

9.53 problem 1908

9.53.1 Solution using Matrix exponential method

9.53.2 Solution using explicit Eigenvalue and Eigenvector method