problem 3(f)

Internal problem ID [6538]
Internal ﬁle name [OUTPUT/5786_Sunday_June_05_2022_03_54_16_PM_17543604/index.tex]

Book: Diﬀerential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 10. Systems of First-Order Equations. Section A. Drill exercises. Page 400
Problem number: 3(f).
ODE order: 1.
ODE degree: 1.

Solve \begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )+y \left (t \right )-z \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \left (t \right )-y \left (t \right )-4 z \left (t \right )\\ z^{\prime }\left (t \right )&=3 x \left (t \right )-y \left (t \right )+z \left (t \right ) \end {align*}

29.10.1 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} -1 & 1 & -1 \\ 2 & -1 & -4 \\ 3 & -1 & 1 \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.

29.10.2 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} -1 & 1 & -1 \\ 2 & -1 & -4 \\ 3 & -1 & 1 \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 & -1 \\ 2 & -1-\lambda & -4 \\ 3 & -1 & 1-\lambda \end {array}\right ]\right ) &= 0 \end {align*}

Which gives the characteristic equation \begin {align*} \lambda ^{3}+\lambda ^{2}-4 \lambda +10&=0 \end {align*}

The roots of the above are the eigenvalues. \begin {align*} \lambda _1 &= -\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\\ \lambda _2 &= \frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\\ \lambda _3 &= \frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}


eigenvalue	algebraic multiplicity	type of eigenvalue

$-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}$	$1$	real eigenvalue

$\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}$	$1$	complex eigenvalue

$\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}$	$1$	complex eigenvalue

Considering the eigenvalue $\lambda _{1} = -\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}$

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} -1 & 1 & -1 \\ 2 & -1 & -4 \\ 3 & -1 & 1 \end {array}\right ] - \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{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 (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}} & 1 & -1 \\ 2 & \frac {\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}} & -4 \\ 3 & -1 & \frac {\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13}{3 \left (154+3 \sqrt {2391}\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 {9 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} R_{1}}{\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}&1&-1&0\\ 0&\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \sqrt {2391}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149}{\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right )}&\frac {-4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+14 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-52}{\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13}&0\\ 0&\frac {-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-13}{\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13}&\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+15 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+2 \sqrt {2391}+60 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+159}{\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right )}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}-\frac {\left (-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-13\right ) \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} R_{2}}{\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \sqrt {2391}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}&1&-1&0\\ 0&\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \sqrt {2391}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149}{\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right )}&\frac {-4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+14 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-52}{\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13}&0\\ 0&0&0&0 \end {array} \right ] \end {align*}

Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} \frac {\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}} & 1 & -1 \\ 0 & \frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \sqrt {2391}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149}{\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right )} & \frac {-4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+14 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-52}{\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13} \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are $\{v_{3}\}$ and the leading variables are $\{v_{1}, v_{2}\}$. Let $v_{3} = t$. Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation $\left \{v_{1} = \frac {3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} t \left (\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+18 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-16 \sqrt {2391}-18 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-765\right )}{\left (\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \sqrt {2391}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149\right ) \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right )}, v_{2} = -\frac {2 t \left (7 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-26 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-6 \sqrt {2391}-308\right )}{\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \sqrt {2391}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149}\right \}$

Hence the solution is \[ \left [\begin {array}{c} \frac {3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} t \left (\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+18 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-16 \sqrt {2391}-18 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-765\right )}{\left (\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \sqrt {2391}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149\right ) \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right )} \\ -\frac {2 t \left (7 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-26 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-6 \sqrt {2391}-308\right )}{\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \sqrt {2391}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} t \left (\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+18 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-16 \sqrt {2391}-18 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-765\right )}{\left (\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \sqrt {2391}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149\right ) \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right )} \\ -\frac {2 t \left (7 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-26 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-6 \sqrt {2391}-308\right )}{\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \sqrt {2391}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149} \\ t \end {array}\right ] \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \left [\begin {array}{c} \frac {3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} t \left (\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+18 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-16 \sqrt {2391}-18 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-765\right )}{\left (\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \sqrt {2391}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149\right ) \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right )} \\ -\frac {2 t \left (7 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-26 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-6 \sqrt {2391}-308\right )}{\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \sqrt {2391}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149} \\ t \end {array}\right ] = t \left [\begin {array}{c} \frac {3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \left (\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+18 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-16 \sqrt {2391}-18 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-765\right )}{\left (\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \sqrt {2391}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149\right ) \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right )} \\ -\frac {2 \left (7 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-26 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-6 \sqrt {2391}-308\right )}{\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \sqrt {2391}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149} \\ 1 \end {array}\right ] \] Let $t = 1$ the eigenvector becomes \[ \left [\begin {array}{c} \frac {3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} t \left (\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+18 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-16 \sqrt {2391}-18 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-765\right )}{\left (\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \sqrt {2391}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149\right ) \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right )} \\ -\frac {2 t \left (7 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-26 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-6 \sqrt {2391}-308\right )}{\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \sqrt {2391}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \left (\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+18 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-16 \sqrt {2391}-18 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-765\right )}{\left (\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \sqrt {2391}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149\right ) \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right )} \\ -\frac {2 \left (7 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-26 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-6 \sqrt {2391}-308\right )}{\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \sqrt {2391}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149} \\ 1 \end {array}\right ] \] Which is normalized to \[ \left [\begin {array}{c} \frac {3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} t \left (\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+18 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-16 \sqrt {2391}-18 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-765\right )}{\left (\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \sqrt {2391}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149\right ) \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right )} \\ -\frac {2 t \left (7 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-26 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-6 \sqrt {2391}-308\right )}{\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \sqrt {2391}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \left (\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+18 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-16 \sqrt {2391}-18 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-765\right )}{\left (\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \sqrt {2391}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149\right ) \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right )} \\ -\frac {2 \left (7 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-26 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-6 \sqrt {2391}-308\right )}{\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \sqrt {2391}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149} \\ 1 \end {array}\right ] \] Considering the eigenvalue $\lambda _{2} = \frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}$

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 & -1 \\ 2 & -1 & -4 \\ 3 & -1 & 1 \end {array}\right ] - \left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\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 (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}+4 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+13}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}} & 1 & -1 \\ 2 & -\frac {\left (1+i \sqrt {3}\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}+4 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+13}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}} & -4 \\ 3 & -1 & -\frac {\left (1+i \sqrt {3}\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}-8 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+13}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\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 \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {2}{3}-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}-\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}&1&-1&0\\ 2&-\frac {2}{3}-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}-\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}&-4&0\\ 3&-1&\frac {4}{3}-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}-\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}-\frac {2 R_{1}}{-\frac {2}{3}-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}-\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\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 (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}+4 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+13}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}}&1&-1&0\\ 0&\frac {\left (-34 i \sqrt {3}-3 i \sqrt {797}+\sqrt {2391}+34\right ) \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149 i \sqrt {3}-12 i \sqrt {797}-4 \sqrt {2391}-8 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-149}{\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \left (13+4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+i \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) \sqrt {3}+\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}\right )}&\frac {-52-28 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+4 i \left (-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+13\right ) \sqrt {3}-4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}}{13+4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+i \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) \sqrt {3}+\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}}&0\\ 3&-1&-\frac {\left (1+i \sqrt {3}\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}-8 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+13}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}+\frac {18 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}} R_{1}}{\left (1+i \sqrt {3}\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}+4 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+13} &\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 (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}+4 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+13}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}}&1&-1&0\\ 0&\frac {\left (-34 i \sqrt {3}-3 i \sqrt {797}+\sqrt {2391}+34\right ) \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149 i \sqrt {3}-12 i \sqrt {797}-4 \sqrt {2391}-8 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-149}{\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \left (13+4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+i \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) \sqrt {3}+\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}\right )}&\frac {-52-28 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+4 i \left (-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+13\right ) \sqrt {3}-4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}}{13+4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+i \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) \sqrt {3}+\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}}&0\\ 0&\frac {-i \sqrt {3}\, \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+13 i \sqrt {3}-\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+14 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-13}{i \sqrt {3}\, \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}+\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+4 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+13}&\frac {-3 i \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}} \sqrt {797}+6 i \sqrt {797}-60 i \sqrt {3}\, \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {797}+159 i \sqrt {3}+2 \sqrt {3}\, \sqrt {797}-30 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+60 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+159}{\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}} \left (i \sqrt {3}\, \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}+\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+4 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+13\right )}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}-\frac {\left (-i \sqrt {3}\, \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+13 i \sqrt {3}-\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+14 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-13\right ) \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \left (13+4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+i \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) \sqrt {3}+\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}\right ) R_{2}}{\left (i \sqrt {3}\, \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}+\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+4 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+13\right ) \left (\left (-34 i \sqrt {3}-3 i \sqrt {797}+\sqrt {2391}+34\right ) \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149 i \sqrt {3}-12 i \sqrt {797}-4 \sqrt {2391}-8 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-149\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 (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}+4 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+13}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}}&1&-1&0\\ 0&\frac {\left (-34 i \sqrt {3}-3 i \sqrt {797}+\sqrt {2391}+34\right ) \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149 i \sqrt {3}-12 i \sqrt {797}-4 \sqrt {2391}-8 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-149}{\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \left (13+4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+i \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) \sqrt {3}+\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}\right )}&\frac {-52-28 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+4 i \left (-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+13\right ) \sqrt {3}-4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}}{13+4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+i \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) \sqrt {3}+\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}}&0\\ 0&0&0&0 \end {array} \right ] \end {align*}

Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} -\frac {\left (1+i \sqrt {3}\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}+4 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+13}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}} & 1 & -1 \\ 0 & \frac {\left (-34 i \sqrt {3}-3 i \sqrt {797}+\sqrt {2391}+34\right ) \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149 i \sqrt {3}-12 i \sqrt {797}-4 \sqrt {2391}-8 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-149}{\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \left (13+4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+i \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) \sqrt {3}+\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}\right )} & \frac {-52-28 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+4 i \left (-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+13\right ) \sqrt {3}-4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}}{13+4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+i \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) \sqrt {3}+\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}} \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] 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 \(\left \{v_{1} = \frac {6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} t \left (18 i \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {3}+\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}-3 i \sqrt {797}\, \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-765 i \sqrt {3}-16 \sqrt {2391}-48 i \sqrt {797}-36 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-18 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-765\right )}{\left (i \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13 i \sqrt {3}+\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right ) \left (3 i \sqrt {797}\, \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+34 i \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {3}-\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+149 i \sqrt {3}+4 \sqrt {2391}+12 i \sqrt {797}+8 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+149\right )}, v_{2} = \frac {4 t \left (13 i \sqrt {3}\, \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-154 i \sqrt {3}-3 \sqrt {3}\, \sqrt {797}-9 i \sqrt {797}-7 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-154\right )}{3 i \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}} \sqrt {797}+34 i \sqrt {3}\, \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {797}+149 i \sqrt {3}+4 \sqrt {3}\, \sqrt {797}+12 i \sqrt {797}+8 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-34 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+149}\right \}\)

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} \frac {6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} t \left (18 \,\operatorname {I} \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {3}+\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}-3 \,\operatorname {I} \sqrt {797}\, \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-765 \,\operatorname {I} \sqrt {3}-16 \sqrt {2391}-48 \,\operatorname {I} \sqrt {797}-36 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-18 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-765\right )}{\left (\operatorname {I} \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13 \,\operatorname {I} \sqrt {3}+\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right ) \left (3 \,\operatorname {I} \sqrt {797}\, \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+34 \,\operatorname {I} \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {3}-\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+149 \,\operatorname {I} \sqrt {3}+4 \sqrt {2391}+12 \,\operatorname {I} \sqrt {797}+8 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+149\right )} \\ \frac {4 t \left (13 \,\operatorname {I} \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}} \sqrt {3}-154 \,\operatorname {I} \sqrt {3}-3 \sqrt {3}\, \sqrt {797}-9 \,\operatorname {I} \sqrt {797}-7 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-154\right )}{3 \,\operatorname {I} \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}} \sqrt {797}+34 \,\operatorname {I} \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}} \sqrt {3}-\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {797}+149 \,\operatorname {I} \sqrt {3}+4 \sqrt {3}\, \sqrt {797}+12 \,\operatorname {I} \sqrt {797}+8 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-34 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+149} \\ t \end {array}\right ] = t \left [\begin {array}{c} \frac {6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \left (18 i \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {3}+\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}-3 i \sqrt {797}\, \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-765 i \sqrt {3}-16 \sqrt {2391}-48 i \sqrt {797}-36 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-18 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-765\right )}{\left (i \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13 i \sqrt {3}+\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right ) \left (3 i \sqrt {797}\, \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+34 i \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {3}-\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+149 i \sqrt {3}+4 \sqrt {2391}+12 i \sqrt {797}+8 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+149\right )} \\ \frac {52 i \sqrt {3}\, \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-616 i \sqrt {3}-12 \sqrt {3}\, \sqrt {797}-36 i \sqrt {797}-28 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-52 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-616}{3 i \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}} \sqrt {797}+34 i \sqrt {3}\, \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {797}+149 i \sqrt {3}+4 \sqrt {3}\, \sqrt {797}+12 i \sqrt {797}+8 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-34 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+149} \\ 1 \end {array}\right ] \] Let $t = 1$ the eigenvector becomes \[ \text {Expression too large to display} \] Which is normalized to \[ \text {Expression too large to display} \] Considering the eigenvalue $\lambda _{3} = \frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}$

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 & -1 \\ 2 & -1 & -4 \\ 3 & -1 & 1 \end {array}\right ] - \left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\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 (i \sqrt {3}-1\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}-4 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-13}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}} & 1 & -1 \\ 2 & \frac {\left (i \sqrt {3}-1\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}-4 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-13}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}} & -4 \\ 3 & -1 & \frac {\left (i \sqrt {3}-1\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}+8 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-13}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\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 \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {2}{3}-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}-\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}&1&-1&0\\ 2&-\frac {2}{3}-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}-\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}&-4&0\\ 3&-1&\frac {4}{3}-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}-\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}-\frac {2 R_{1}}{-\frac {2}{3}-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}-\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\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 (i \sqrt {3}-1\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}-4 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-13}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}}&1&-1&0\\ 0&\frac {-\left (34 i \sqrt {3}+3 i \sqrt {797}+\sqrt {2391}+34\right ) \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149 i \sqrt {3}-12 i \sqrt {797}+4 \sqrt {2391}+8 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+149}{\left (-13-4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+i \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) \sqrt {3}-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}\right ) \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}&\frac {52+28 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+4 i \left (-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+13\right ) \sqrt {3}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}}{-13-4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+i \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) \sqrt {3}-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}}&0\\ 3&-1&\frac {\left (i \sqrt {3}-1\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}+8 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-13}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}-\frac {18 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}} R_{1}}{\left (i \sqrt {3}-1\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}-4 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-13} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {\left (i \sqrt {3}-1\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}-4 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-13}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}}&1&-1&0\\ 0&\frac {-\left (34 i \sqrt {3}+3 i \sqrt {797}+\sqrt {2391}+34\right ) \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149 i \sqrt {3}-12 i \sqrt {797}+4 \sqrt {2391}+8 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+149}{\left (-13-4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+i \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) \sqrt {3}-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}\right ) \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}&\frac {52+28 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+4 i \left (-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+13\right ) \sqrt {3}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}}{-13-4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+i \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) \sqrt {3}-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}}&0\\ 0&\frac {-i \sqrt {3}\, \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+13 i \sqrt {3}+\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-14 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+13}{i \sqrt {3}\, \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}-\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-4 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-13}&\frac {-\left (60 i \sqrt {3}+3 i \sqrt {797}+\sqrt {2391}+60\right ) \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+159 i \sqrt {3}+6 i \sqrt {797}-2 \sqrt {2391}+30 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-159}{\left (-13-4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+i \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) \sqrt {3}-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}\right ) \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}-\frac {\left (-i \sqrt {3}\, \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+13 i \sqrt {3}+\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-14 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+13\right ) \left (-13-4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+i \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) \sqrt {3}-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}\right ) \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} R_{2}}{\left (i \sqrt {3}\, \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}-\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-4 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-13\right ) \left (-\left (34 i \sqrt {3}+3 i \sqrt {797}+\sqrt {2391}+34\right ) \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149 i \sqrt {3}-12 i \sqrt {797}+4 \sqrt {2391}+8 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+149\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {\left (i \sqrt {3}-1\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}-4 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-13}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}}&1&-1&0\\ 0&\frac {-\left (34 i \sqrt {3}+3 i \sqrt {797}+\sqrt {2391}+34\right ) \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149 i \sqrt {3}-12 i \sqrt {797}+4 \sqrt {2391}+8 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+149}{\left (-13-4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+i \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) \sqrt {3}-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}\right ) \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}&\frac {52+28 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+4 i \left (-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+13\right ) \sqrt {3}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}}{-13-4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+i \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) \sqrt {3}-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}}&0\\ 0&0&0&0 \end {array} \right ] \end {align*}

Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} \frac {\left (i \sqrt {3}-1\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13 i \sqrt {3}-4 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-13}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}} & 1 & -1 \\ 0 & \frac {-\left (34 i \sqrt {3}+3 i \sqrt {797}+\sqrt {2391}+34\right ) \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149 i \sqrt {3}-12 i \sqrt {797}+4 \sqrt {2391}+8 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+149}{\left (-13-4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+i \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) \sqrt {3}-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}\right ) \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}} & \frac {52+28 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+4 i \left (-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+13\right ) \sqrt {3}+4 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}}{-13-4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+i \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) \sqrt {3}-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}} \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] 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 \(\left \{v_{1} = -\frac {6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} t \left (18 i \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {3}-\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}-3 i \sqrt {797}\, \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-765 i \sqrt {3}+16 \sqrt {2391}-48 i \sqrt {797}+36 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+18 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+765\right )}{\left (i \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13 i \sqrt {3}-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-13\right ) \left (34 i \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {3}+\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+3 i \sqrt {797}\, \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+149 i \sqrt {3}-4 \sqrt {2391}+12 i \sqrt {797}-8 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149\right )}, v_{2} = \frac {4 t \left (13 i \sqrt {3}\, \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-154 i \sqrt {3}+3 \sqrt {3}\, \sqrt {797}-9 i \sqrt {797}+7 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+13 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+154\right )}{34 i \sqrt {3}\, \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {797}+3 i \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}} \sqrt {797}+149 i \sqrt {3}-4 \sqrt {3}\, \sqrt {797}+12 i \sqrt {797}-8 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+34 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-149}\right \}\)

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} -\frac {6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} t \left (18 \,\operatorname {I} \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {3}-\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}-3 \,\operatorname {I} \sqrt {797}\, \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-765 \,\operatorname {I} \sqrt {3}+16 \sqrt {2391}-48 \,\operatorname {I} \sqrt {797}+36 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+18 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+765\right )}{\left (\operatorname {I} \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13 \,\operatorname {I} \sqrt {3}-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-13\right ) \left (34 \,\operatorname {I} \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {3}+\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+3 \,\operatorname {I} \sqrt {797}\, \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+149 \,\operatorname {I} \sqrt {3}-4 \sqrt {2391}+12 \,\operatorname {I} \sqrt {797}-8 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149\right )} \\ \frac {4 t \left (13 \,\operatorname {I} \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}} \sqrt {3}-154 \,\operatorname {I} \sqrt {3}+3 \sqrt {3}\, \sqrt {797}-9 \,\operatorname {I} \sqrt {797}+7 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+13 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+154\right )}{34 \,\operatorname {I} \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}} \sqrt {3}+\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {797}+3 \,\operatorname {I} \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}} \sqrt {797}+149 \,\operatorname {I} \sqrt {3}-4 \sqrt {3}\, \sqrt {797}+12 \,\operatorname {I} \sqrt {797}-8 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+34 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-149} \\ t \end {array}\right ] = t \left [\begin {array}{c} -\frac {6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \left (18 i \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {3}-\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}-3 i \sqrt {797}\, \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-765 i \sqrt {3}+16 \sqrt {2391}-48 i \sqrt {797}+36 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+18 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+765\right )}{\left (i \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13 i \sqrt {3}-\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-4 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-13\right ) \left (34 i \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {3}+\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sqrt {2391}+3 i \sqrt {797}\, \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+149 i \sqrt {3}-4 \sqrt {2391}+12 i \sqrt {797}-8 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+34 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}-149\right )} \\ \frac {52 i \sqrt {3}\, \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-616 i \sqrt {3}+12 \sqrt {3}\, \sqrt {797}-36 i \sqrt {797}+28 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+52 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+616}{34 i \sqrt {3}\, \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {797}+3 i \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}} \sqrt {797}+149 i \sqrt {3}-4 \sqrt {3}\, \sqrt {797}+12 i \sqrt {797}-8 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+34 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-149} \\ 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

$-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}$	$1$	$1$	No	\(\left [\begin {array}{c} -\frac {7 \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {14}{3}\right )}{\left (-\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {91}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}\right ) \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )} \\ \frac {\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}-32}{-\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {91}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}} \\ 1 \end {array}\right ]\)

$\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}$	$1$	$1$	No	\(\left [\begin {array}{c} -\frac {7 \left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {14}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {91}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {7 i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}+\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-32}{\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {91}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {7 i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}+\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\)

$\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}$	$1$	$1$	No	\(\left [\begin {array}{c} -\frac {7 \left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {14}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {91}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {7 i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}+\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-32}{\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {91}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {7 i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}+\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\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. Since eigenvalue $-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}$ is real and distinct then the corresponding eigenvector solution is \begin {align*} \vec {x}_{1}(t) &= \vec {v}_{1} e^{\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t}\\ &= \left [\begin {array}{c} -\frac {7 \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {14}{3}\right )}{\left (-\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {91}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}\right ) \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )} \\ \frac {\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}-32}{-\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {91}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}} \\ 1 \end {array}\right ] e^{\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{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 {7 \,{\mathrm e}^{\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t} \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {14}{3}\right )}{\left (-\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {91}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}\right ) \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )} \\ \frac {{\mathrm e}^{\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t} \left (\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}-32\right )}{-\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {91}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}} \\ {\mathrm e}^{\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{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*}

29.10.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [x^{\prime }\left (t \right )=-x \left (t \right )+y \left (t \right )-z \left (t \right ), y^{\prime }\left (t \right )=2 x \left (t \right )-y \left (t \right )-4 z \left (t \right ), z^{\prime }\left (t \right )=3 x \left (t \right )-y \left (t \right )+z \left (t \right )\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 & -1 \\ 2 & -1 & -4 \\ 3 & -1 & 1 \end {array}\right ]\cdot {\moverset {\rightarrow }{x}}\left (t \right )+\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}^{\prime }\left (t \right )=\left [\begin {array}{ccc} -1 & 1 & -1 \\ 2 & -1 & -4 \\ 3 & -1 & 1 \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 & -1 \\ 2 & -1 & -4 \\ 3 & -1 & 1 \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 (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}, \left [\begin {array}{c} -\frac {7 \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {14}{3}\right )}{\left (-\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {91}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}\right ) \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )} \\ \frac {\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}-32}{-\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {91}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}} \\ 1 \end {array}\right ]\right ], \left [\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} -\frac {7 \left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {14}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {91}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {7 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}+\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-32}{\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {91}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {7 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}+\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\right ], \left [\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} -\frac {7 \left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {14}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {91}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {7 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}+\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-32}{\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {91}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {7 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}+\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}, \left [\begin {array}{c} -\frac {7 \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {14}{3}\right )}{\left (-\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {91}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}\right ) \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )} \\ \frac {\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}-32}{-\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {91}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\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 (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t}\cdot \left [\begin {array}{c} -\frac {7 \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {14}{3}\right )}{\left (-\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {91}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}\right ) \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )} \\ \frac {\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}-32}{-\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {91}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} -\frac {7 \left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {14}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {91}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {7 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}+\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-32}{\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {91}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {7 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}+\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}\cdot \left [\begin {array}{c} -\frac {7 \left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {14}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {91}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {7 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}+\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-32}{\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {91}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {7 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}+\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Use Euler identity to write solution in terms of}\hspace {3pt} \sin \hspace {3pt}\textrm {and}\hspace {3pt} \cos \\ {} & {} & {\mathrm e}^{\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t}\cdot \left (\cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) t}{2}\right )\right )\cdot \left [\begin {array}{c} -\frac {7 \left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {14}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {91}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {7 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}+\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-32}{\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {91}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {7 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}+\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t}\cdot \left [\begin {array}{c} -\frac {7 \left (\cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) t}{2}\right )\right ) \left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {14}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {91}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {7 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}+\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {\left (\cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) t}{2}\right )\right ) \left (\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-32\right )}{\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {91}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {7 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}+\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) t}{2}\right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{x}}_{2}\left (t \right )={\mathrm e}^{\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t}\cdot \left [\begin {array}{c} -\frac {63 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}} \left (\sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {8}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+21 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-\left (154+3 \sqrt {2391}\right )^{\frac {8}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-7 \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+616 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+1190 \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+13496 \left (154+3 \sqrt {2391}\right )^{\frac {4}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-47489 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+11449242 \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-46137 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+235620 \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {2391}-23491 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-3933846 \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-15379 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-82026 \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) \sqrt {2391}\right )}{2 \left (\left (154+3 \sqrt {2391}\right )^{\frac {8}{3}}+19 \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}}+133 \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}}-3773 \left (154+3 \sqrt {2391}\right )^{\frac {4}{3}}+55432 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+15131907+41743 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+308259 \sqrt {2391}\right ) \left (\left (154+3 \sqrt {2391}\right )^{\frac {4}{3}}-9 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+26 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+6 \sqrt {2391}+477\right )} \\ -\frac {21 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-2 \left (154+3 \sqrt {2391}\right )^{\frac {8}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-17 \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+4893 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+5467 \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+16576 \left (154+3 \sqrt {2391}\right )^{\frac {4}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-49686 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+3503304 \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-46137 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+80829 \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {2391}-25688 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+4012092 \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-37349 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+72765 \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) \sqrt {2391}}{2 \left (\left (154+3 \sqrt {2391}\right )^{\frac {8}{3}}+19 \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}}+133 \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}}-3773 \left (154+3 \sqrt {2391}\right )^{\frac {4}{3}}+55432 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+15131907+41743 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+308259 \sqrt {2391}\right )} \\ \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) t}{2}\right ) \end {array}\right ], {\moverset {\rightarrow }{x}}_{3}\left (t \right )={\mathrm e}^{\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t}\cdot \left [\begin {array}{c} \frac {63 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}} \left (\sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {8}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+21 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+\left (154+3 \sqrt {2391}\right )^{\frac {8}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+7 \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+616 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-1190 \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-13496 \left (154+3 \sqrt {2391}\right )^{\frac {4}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-47489 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+11449242 \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-46137 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+235620 \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {2391}+23491 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+3933846 \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+15379 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+82026 \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) \sqrt {2391}\right )}{2 \left (\left (154+3 \sqrt {2391}\right )^{\frac {8}{3}}+19 \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}}+133 \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}}-3773 \left (154+3 \sqrt {2391}\right )^{\frac {4}{3}}+55432 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+15131907+41743 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+308259 \sqrt {2391}\right ) \left (\left (154+3 \sqrt {2391}\right )^{\frac {4}{3}}-9 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+26 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+6 \sqrt {2391}+477\right )} \\ \frac {21 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+2 \left (154+3 \sqrt {2391}\right )^{\frac {8}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+17 \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+4893 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-5467 \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-16576 \left (154+3 \sqrt {2391}\right )^{\frac {4}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-49686 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+3503304 \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-46137 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+80829 \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {2391}+25688 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-4012092 \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+37349 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-72765 \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) \sqrt {2391}}{2 \left (\left (154+3 \sqrt {2391}\right )^{\frac {8}{3}}+19 \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}}+133 \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}}-3773 \left (154+3 \sqrt {2391}\right )^{\frac {4}{3}}+55432 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+15131907+41743 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+308259 \sqrt {2391}\right )} \\ -\sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) t}{2}\right ) \end {array}\right ]\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}\left (t \right )+c_{3} {\moverset {\rightarrow }{x}}_{3}\left (t \right ) \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}=c_{1} {\mathrm e}^{\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t}\cdot \left [\begin {array}{c} -\frac {7 \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {14}{3}\right )}{\left (-\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {91}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}\right ) \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )} \\ \frac {\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}-32}{-\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {91}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}} \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t}\cdot \left [\begin {array}{c} -\frac {63 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}} \left (\sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {8}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+21 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-\left (154+3 \sqrt {2391}\right )^{\frac {8}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-7 \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+616 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+1190 \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+13496 \left (154+3 \sqrt {2391}\right )^{\frac {4}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-47489 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+11449242 \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-46137 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+235620 \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {2391}-23491 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-3933846 \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-15379 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-82026 \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) \sqrt {2391}\right )}{2 \left (\left (154+3 \sqrt {2391}\right )^{\frac {8}{3}}+19 \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}}+133 \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}}-3773 \left (154+3 \sqrt {2391}\right )^{\frac {4}{3}}+55432 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+15131907+41743 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+308259 \sqrt {2391}\right ) \left (\left (154+3 \sqrt {2391}\right )^{\frac {4}{3}}-9 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+26 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+6 \sqrt {2391}+477\right )} \\ -\frac {21 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-2 \left (154+3 \sqrt {2391}\right )^{\frac {8}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-17 \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+4893 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+5467 \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+16576 \left (154+3 \sqrt {2391}\right )^{\frac {4}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-49686 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+3503304 \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-46137 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+80829 \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {2391}-25688 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+4012092 \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-37349 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+72765 \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) \sqrt {2391}}{2 \left (\left (154+3 \sqrt {2391}\right )^{\frac {8}{3}}+19 \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}}+133 \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}}-3773 \left (154+3 \sqrt {2391}\right )^{\frac {4}{3}}+55432 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+15131907+41743 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+308259 \sqrt {2391}\right )} \\ \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) t}{2}\right ) \end {array}\right ]+c_{3} {\mathrm e}^{\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t}\cdot \left [\begin {array}{c} \frac {63 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}} \left (\sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {8}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+21 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+\left (154+3 \sqrt {2391}\right )^{\frac {8}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+7 \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+616 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-1190 \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-13496 \left (154+3 \sqrt {2391}\right )^{\frac {4}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-47489 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+11449242 \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-46137 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+235620 \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {2391}+23491 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+3933846 \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+15379 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+82026 \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) \sqrt {2391}\right )}{2 \left (\left (154+3 \sqrt {2391}\right )^{\frac {8}{3}}+19 \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}}+133 \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}}-3773 \left (154+3 \sqrt {2391}\right )^{\frac {4}{3}}+55432 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+15131907+41743 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+308259 \sqrt {2391}\right ) \left (\left (154+3 \sqrt {2391}\right )^{\frac {4}{3}}-9 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+26 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+6 \sqrt {2391}+477\right )} \\ \frac {21 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+2 \left (154+3 \sqrt {2391}\right )^{\frac {8}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+17 \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+4893 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-5467 \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-16576 \left (154+3 \sqrt {2391}\right )^{\frac {4}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-49686 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+3503304 \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-46137 \sqrt {3}\, \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+80829 \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {2391}+25688 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-4012092 \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+37349 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )-72765 \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) \sqrt {2391}}{2 \left (\left (154+3 \sqrt {2391}\right )^{\frac {8}{3}}+19 \left (154+3 \sqrt {2391}\right )^{\frac {7}{3}}+133 \left (154+3 \sqrt {2391}\right )^{\frac {5}{3}}-3773 \left (154+3 \sqrt {2391}\right )^{\frac {4}{3}}+55432 \left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+15131907+41743 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+308259 \sqrt {2391}\right )} \\ -\sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right ) t}{2}\right ) \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 {16632 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}} \left (\left (\left (\left (\left (-\frac {1987 \sqrt {3}\, c_{2}}{16632}-\frac {1987 c_{3}}{5544}\right ) \sqrt {797}-\frac {4613 c_{2}}{792}-\frac {4613 \sqrt {3}\, c_{3}}{792}\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+c_{2} \left (\frac {45235}{924}+\sqrt {3}\, \sqrt {797}\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+\left (\frac {64639 \sqrt {3}\, c_{2}}{16632}-\frac {64639 c_{3}}{5544}\right ) \sqrt {797}+\frac {150439 c_{2}}{792}-\frac {150439 \sqrt {3}\, c_{3}}{792}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}}\right )+\sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}}\right ) \left (\left (\left (\frac {1987 c_{2}}{5544}-\frac {1987 \sqrt {3}\, c_{3}}{16632}\right ) \sqrt {797}+\frac {4613 \sqrt {3}\, c_{2}}{792}-\frac {4613 c_{3}}{792}\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+c_{3} \left (\frac {45235}{924}+\sqrt {3}\, \sqrt {797}\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+\left (\frac {64639 c_{2}}{5544}+\frac {64639 \sqrt {3}\, c_{3}}{16632}\right ) \sqrt {797}+\frac {150439 \sqrt {3}\, c_{2}}{792}+\frac {150439 c_{3}}{792}\right )\right ) {\mathrm e}^{\frac {\left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}}+\left (\left (\frac {1987 \sqrt {3}\, \sqrt {797}}{8316}+\frac {4613}{396}\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+\left (\frac {45235}{924}+\sqrt {3}\, \sqrt {797}\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-\frac {150439}{396}-\frac {64639 \sqrt {3}\, \sqrt {797}}{8316}\right ) {\mathrm e}^{-\frac {\left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right ) t}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}} c_{1} \right )}{11676042 \sqrt {3}\, \sqrt {797}+570949764} \\ \frac {\left (\left (\left (\left (-507 \sqrt {3}\, c_{2} -1521 c_{3} \right ) \sqrt {797}-26026 \sqrt {3}\, c_{3} -26026 c_{2} \right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+\left (\left (-924 \sqrt {3}\, c_{2} +2772 c_{3} \right ) \sqrt {797}+45235 \sqrt {3}\, c_{3} -45235 c_{2} \right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+1848 c_{2} \sqrt {797}\, \sqrt {3}+90470 c_{2} \right ) \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}}\right )+26026 \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}}\right ) \left (\left (\left (-\frac {3 \sqrt {3}\, c_{3}}{154}+\frac {9 c_{2}}{154}\right ) \sqrt {797}+\sqrt {3}\, c_{2} -c_{3} \right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+\left (\left (-\frac {6 \sqrt {3}\, c_{3}}{169}-\frac {18 c_{2}}{169}\right ) \sqrt {797}-\frac {45235 \sqrt {3}\, c_{2}}{26026}-\frac {45235 c_{3}}{26026}\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+\frac {12 c_{3} \left (\frac {45235}{924}+\sqrt {3}\, \sqrt {797}\right )}{169}\right )\right ) {\mathrm e}^{\frac {\left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}}+1014 \left (\left (\sqrt {3}\, \sqrt {797}+\frac {154}{3}\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+\frac {308 \left (\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+1\right ) \left (\frac {45235}{924}+\sqrt {3}\, \sqrt {797}\right )}{169}\right ) {\mathrm e}^{-\frac {\left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right ) t}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}} c_{1}}{38808 \sqrt {3}\, \sqrt {797}+1899870} \\ {\mathrm e}^{-\frac {\left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right ) t}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}} c_{1} +{\mathrm e}^{\frac {\left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}}\right ) c_{2} +{\mathrm e}^{\frac {\left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}}\right ) c_{3} \end {array}\right ] \\ \bullet & {} & \textrm {Solution to the system of ODEs}\hspace {3pt} \\ {} & {} & \left \{x \left (t \right )=-\frac {16632 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}} \left (\left (\left (\left (\left (-\frac {1987 \sqrt {3}\, c_{2}}{16632}-\frac {1987 c_{3}}{5544}\right ) \sqrt {797}-\frac {4613 c_{2}}{792}-\frac {4613 \sqrt {3}\, c_{3}}{792}\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+c_{2} \left (\frac {45235}{924}+\sqrt {3}\, \sqrt {797}\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+\left (\frac {64639 \sqrt {3}\, c_{2}}{16632}-\frac {64639 c_{3}}{5544}\right ) \sqrt {797}+\frac {150439 c_{2}}{792}-\frac {150439 \sqrt {3}\, c_{3}}{792}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}}\right )+\sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}}\right ) \left (\left (\left (\frac {1987 c_{2}}{5544}-\frac {1987 \sqrt {3}\, c_{3}}{16632}\right ) \sqrt {797}+\frac {4613 \sqrt {3}\, c_{2}}{792}-\frac {4613 c_{3}}{792}\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+c_{3} \left (\frac {45235}{924}+\sqrt {3}\, \sqrt {797}\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+\left (\frac {64639 c_{2}}{5544}+\frac {64639 \sqrt {3}\, c_{3}}{16632}\right ) \sqrt {797}+\frac {150439 \sqrt {3}\, c_{2}}{792}+\frac {150439 c_{3}}{792}\right )\right ) {\mathrm e}^{\frac {\left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}}+\left (\left (\frac {1987 \sqrt {3}\, \sqrt {797}}{8316}+\frac {4613}{396}\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+\left (\frac {45235}{924}+\sqrt {3}\, \sqrt {797}\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}-\frac {150439}{396}-\frac {64639 \sqrt {3}\, \sqrt {797}}{8316}\right ) {\mathrm e}^{-\frac {\left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right ) t}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}} c_{1} \right )}{11676042 \sqrt {3}\, \sqrt {797}+570949764}, y \left (t \right )=\frac {\left (\left (\left (\left (-507 \sqrt {3}\, c_{2} -1521 c_{3} \right ) \sqrt {797}-26026 \sqrt {3}\, c_{3} -26026 c_{2} \right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+\left (\left (-924 \sqrt {3}\, c_{2} +2772 c_{3} \right ) \sqrt {797}+45235 \sqrt {3}\, c_{3} -45235 c_{2} \right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+1848 c_{2} \sqrt {797}\, \sqrt {3}+90470 c_{2} \right ) \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}}\right )+26026 \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}}\right ) \left (\left (\left (-\frac {3 \sqrt {3}\, c_{3}}{154}+\frac {9 c_{2}}{154}\right ) \sqrt {797}+\sqrt {3}\, c_{2} -c_{3} \right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+\left (\left (-\frac {6 \sqrt {3}\, c_{3}}{169}-\frac {18 c_{2}}{169}\right ) \sqrt {797}-\frac {45235 \sqrt {3}\, c_{2}}{26026}-\frac {45235 c_{3}}{26026}\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+\frac {12 c_{3} \left (\frac {45235}{924}+\sqrt {3}\, \sqrt {797}\right )}{169}\right )\right ) {\mathrm e}^{\frac {\left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}}+1014 \left (\left (\sqrt {3}\, \sqrt {797}+\frac {154}{3}\right ) \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}+\frac {308 \left (\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}+1\right ) \left (\frac {45235}{924}+\sqrt {3}\, \sqrt {797}\right )}{169}\right ) {\mathrm e}^{-\frac {\left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right ) t}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}} c_{1}}{38808 \sqrt {3}\, \sqrt {797}+1899870}, z \left (t \right )={\mathrm e}^{-\frac {\left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right ) t}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}} c_{1} +{\mathrm e}^{\frac {\left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}}\right ) c_{2} +{\mathrm e}^{\frac {\left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {3}\, \sqrt {797}\right )^{\frac {1}{3}}}\right ) c_{3} \right \} \end {array} \]

\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{\frac {\left (13+\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+c_{3} {\mathrm e}^{\frac {\left (13+\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-2 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}-13\right ) t}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )+c_{1} {\mathrm e}^{-\frac {\left (\left (154+3 \sqrt {2391}\right )^{\frac {2}{3}}+\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}+13\right ) t}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}} \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align*}

\begin{align*} x(t)\to c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ]-c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}+5 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-5 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ] \\ y(t)\to 2 c_1 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-7 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ]-2 c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {2 \text {$\#$1} e^{\text {$\#$1} t}+3 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+2 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ] \\ z(t)\to -c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-2 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {3 \text {$\#$1} e^{\text {$\#$1} t}+e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2-4 \text {$\#$1}+10\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+2 \text {$\#$1} e^{\text {$\#$1} t}-e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}-4}\&\right ] \\ \end{align*}


	multiplicity

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

\(-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\)	\(1\)	\(1\)	No	\(\left [\begin {array}{c} -\frac {7 \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {14}{3}\right )}{\left (-\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {91}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}\right ) \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )} \\ \frac {\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}-32}{-\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {91}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}-\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}} \\ 1 \end {array}\right ]\)

\(\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\)	\(1\)	\(1\)	No	\(\left [\begin {array}{c} -\frac {7 \left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {14}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {91}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {7 i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}+\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-32}{\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {91}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {7 i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}+\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\)

\(\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\)	\(1\)	\(1\)	No	\(\left [\begin {array}{c} -\frac {7 \left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {14}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {91}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {7 i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}+\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-32}{\frac {7 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {91}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {7 i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}+\left (\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{6}+\frac {13}{6 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}{3}+\frac {13}{3 \left (154+3 \sqrt {2391}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\)

29.10 problem 3(f)

29.10.1 Solution using Matrix exponential method

29.10.2 Solution using explicit Eigenvalue and Eigenvector method

29.10.3 Maple step by step solution