problem 4

Internal problem ID [6714]
Internal ﬁle name [OUTPUT/5962_Sunday_June_05_2022_04_04_45_PM_83937430/index.tex]

Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section: CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS. EXERCISES 8.1. Page 332
Problem number: 4.
ODE order: 1.
ODE degree: 1.

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

9.4.1 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 } \\ z^{\prime }\left (t \right ) \end {array}\right ] &= \left [\begin {array}{ccc} 1 & -1 & 0 \\ 1 & 0 & 2 \\ -1 & 0 & 1 \end {array}\right ]\, \left [\begin {array}{c} x \left (t \right ) \\ y \\ z \left (t \right ) \end {array}\right ] \end {align*}

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

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

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

9.4.2 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 & 0 \\ 1 & 0 & 2 \\ -1 & 0 & 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 & 0 \\ 1 & -\lambda & 2 \\ -1 & 0 & 1-\lambda \end {array}\right ]\right ) &= 0 \end {align*}

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

The roots of the above are the eigenvalues. \begin {align*} \lambda _1 &= \frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\\ \lambda _2 &= -\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\\ \lambda _3 &= -\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}


eigenvalue	algebraic multiplicity	type of eigenvalue

$-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}$	$1$	complex eigenvalue

$-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}$	$1$	complex eigenvalue

$\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}$	$1$	real eigenvalue

Considering the eigenvalue $\lambda _{1} = \frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}$

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 & 0 \\ 1 & 0 & 2 \\ -1 & 0 & 1 \end {array}\right ] - \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{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 (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+8}{6 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}} & -1 & 0 \\ 1 & \frac {-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+8}{6 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}} & 2 \\ -1 & 0 & \frac {-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+8}{6 \left (244+12 \sqrt {417}\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 {6 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} R_{1}}{-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+8} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+8}{6 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}&-1&0&0\\ 0&\frac {2 \left (\sqrt {417}+19\right ) \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+4 \sqrt {417}+2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+92}{\left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \left (-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+8\right )}&2&0\\ 0&\frac {6 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-8}&\frac {-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+8}{6 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}-\frac {3 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \left (-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+8\right ) R_{2}}{\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-8\right ) \left (\left (\sqrt {417}+19\right ) \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+2 \sqrt {417}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+46\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+8}{6 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}&-1&0&0\\ 0&\frac {2 \left (\sqrt {417}+19\right ) \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+4 \sqrt {417}+2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+92}{\left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \left (-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+8\right )}&2&0\\ 0&0&0&0 \end {array} \right ] \end {align*}

Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} \frac {-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+8}{6 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}} & -1 & 0 \\ 0 & \frac {2 \left (\sqrt {417}+19\right ) \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+4 \sqrt {417}+2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+92}{\left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \left (-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+8\right )} & 2 \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] 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 $\left \{v_{1} = \frac {\left (3 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {417}-61 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-16 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+32\right ) t}{96}, v_{2} = \frac {2 t \left (-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+6 \sqrt {417}-4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+122\right )}{\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \sqrt {417}+19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+46}\right \}$

Hence the solution is \[ \left [\begin {array}{c} \frac {\left (3 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {417}-61 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-16 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+32\right ) t}{96} \\ \frac {2 t \left (-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+6 \sqrt {417}-4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+122\right )}{\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \sqrt {417}+19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+46} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {\left (3 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {417}-61 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-16 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+32\right ) t}{96} \\ \frac {2 t \left (-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+6 \sqrt {417}-4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+122\right )}{\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \sqrt {417}+19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+46} \\ t \end {array}\right ] \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \left [\begin {array}{c} \frac {\left (3 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {417}-61 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-16 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+32\right ) t}{96} \\ \frac {2 t \left (-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+6 \sqrt {417}-4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+122\right )}{\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \sqrt {417}+19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+46} \\ t \end {array}\right ] = t \left [\begin {array}{c} \frac {\left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {417}}{32}-\frac {61 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}}{96}-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {1}{3} \\ \frac {-2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+12 \sqrt {417}-8 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+244}{\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \sqrt {417}+19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+46} \\ 1 \end {array}\right ] \] Let $t = 1$ the eigenvector becomes \[ \left [\begin {array}{c} \frac {\left (3 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {417}-61 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-16 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+32\right ) t}{96} \\ \frac {2 t \left (-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+6 \sqrt {417}-4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+122\right )}{\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \sqrt {417}+19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+46} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {\left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {417}}{32}-\frac {61 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}}{96}-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {1}{3} \\ \frac {-2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+12 \sqrt {417}-8 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+244}{\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \sqrt {417}+19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+46} \\ 1 \end {array}\right ] \] Which is normalized to \[ \left [\begin {array}{c} \frac {\left (3 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {417}-61 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-16 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+32\right ) t}{96} \\ \frac {2 t \left (-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+6 \sqrt {417}-4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+122\right )}{\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \sqrt {417}+19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+46} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {\left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {417}}{32}-\frac {61 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}}{96}-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {1}{3} \\ \frac {-2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+12 \sqrt {417}-8 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+244}{\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \sqrt {417}+19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+46} \\ 1 \end {array}\right ] \] Considering the eigenvalue $\lambda _{2} = -\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}$

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 & 0 \\ 1 & 0 & 2 \\ -1 & 0 & 1 \end {array}\right ] - \left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\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 {i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}+8 i \sqrt {3}+\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-8}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}} & -1 & 0 \\ 1 & \frac {i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}+\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+8 i \sqrt {3}-8 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-8}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}} & 2 \\ -1 & 0 & \frac {i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}+8 i \sqrt {3}+\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-8}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\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 \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {1}{3}+\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}&-1&0&0\\ 1&-\frac {2}{3}+\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}&2&0\\ -1&0&\frac {1}{3}+\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}-\frac {R_{1}}{\frac {1}{3}+\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\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 {i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}+8 i \sqrt {3}+\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-8}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}&-1&0&0\\ 0&\frac {2 \left (19 i \sqrt {3}+3 i \sqrt {139}-\sqrt {417}-19\right ) \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-92 i \sqrt {3}-12 i \sqrt {139}-4 \sqrt {417}+4 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-92}{\left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \left (-8+4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+i \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) \sqrt {3}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}\right )}&2&0\\ -1&0&\frac {i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}+8 i \sqrt {3}+\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-8}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}+\frac {12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}} R_{1}}{i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}+8 i \sqrt {3}+\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-8} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}+8 i \sqrt {3}+\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-8}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}&-1&0&0\\ 0&\frac {2 \left (19 i \sqrt {3}+3 i \sqrt {139}-\sqrt {417}-19\right ) \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-92 i \sqrt {3}-12 i \sqrt {139}-4 \sqrt {417}+4 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-92}{\left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \left (-8+4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+i \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) \sqrt {3}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}\right )}&2&0\\ 0&-\frac {12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{i \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}+8 i \sqrt {3}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-8}&\frac {i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}+8 i \sqrt {3}+\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-8}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}+\frac {6 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \left (-8+4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+i \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) \sqrt {3}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}\right ) R_{2}}{\left (i \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}+8 i \sqrt {3}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-8\right ) \left (\left (19 i \sqrt {3}+3 i \sqrt {139}-\sqrt {417}-19\right ) \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-46 i \sqrt {3}-6 i \sqrt {139}-2 \sqrt {417}+2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-46\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}+8 i \sqrt {3}+\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-8}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}&-1&0&0\\ 0&\frac {2 \left (19 i \sqrt {3}+3 i \sqrt {139}-\sqrt {417}-19\right ) \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-92 i \sqrt {3}-12 i \sqrt {139}-4 \sqrt {417}+4 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-92}{\left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \left (-8+4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+i \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) \sqrt {3}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}\right )}&2&0\\ 0&0&0&0 \end {array} \right ] \end {align*}

Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} \frac {i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}+8 i \sqrt {3}+\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-8}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}} & -1 & 0 \\ 0 & \frac {2 \left (19 i \sqrt {3}+3 i \sqrt {139}-\sqrt {417}-19\right ) \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-92 i \sqrt {3}-12 i \sqrt {139}-4 \sqrt {417}+4 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-92}{\left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \left (-8+4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+i \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) \sqrt {3}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}\right )} & 2 \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] 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 $\left \{v_{1} = -\frac {\left (61 i \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}-9 i \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {139}+3 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {417}-16 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-61 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-16 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-64\right ) t}{192}, v_{2} = -\frac {4 t \left (2 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}+61 i \sqrt {3}+3 \sqrt {417}+9 i \sqrt {139}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+61\right )}{-\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+3 i \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+19 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-2 \sqrt {417}-6 i \sqrt {139}-46 i \sqrt {3}+2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-46}\right \}$

Hence the solution is \[ \left [\begin {array}{c} -\frac {\left (61 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}-9 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {139}+3 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {417}-16 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-61 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-16 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-64\right ) t}{192} \\ -\frac {4 t \left (2 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}+61 \,\operatorname {I} \sqrt {3}+3 \sqrt {417}+9 \,\operatorname {I} \sqrt {139}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+61\right )}{-\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+3 \,\operatorname {I} \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+19 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-2 \sqrt {417}-6 \,\operatorname {I} \sqrt {139}-46 \,\operatorname {I} \sqrt {3}+2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-46} \\ t \end {array}\right ] = \left [\begin {array}{c} -\frac {\left (61 i \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}-9 i \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {139}+3 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {417}-16 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-61 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-16 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-64\right ) t}{192} \\ -\frac {4 t \left (2 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}+61 i \sqrt {3}+3 \sqrt {417}+9 i \sqrt {139}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+61\right )}{-\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+3 i \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+19 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-2 \sqrt {417}-6 i \sqrt {139}-46 i \sqrt {3}+2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-46} \\ 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 {\left (61 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}-9 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {139}+3 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {417}-16 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-61 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-16 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-64\right ) t}{192} \\ -\frac {4 t \left (2 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}+61 \,\operatorname {I} \sqrt {3}+3 \sqrt {417}+9 \,\operatorname {I} \sqrt {139}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+61\right )}{-\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+3 \,\operatorname {I} \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+19 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-2 \sqrt {417}-6 \,\operatorname {I} \sqrt {139}-46 \,\operatorname {I} \sqrt {3}+2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-46} \\ t \end {array}\right ] = t \left [\begin {array}{c} -\frac {61 i \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}}{192}+\frac {3 i \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {139}}{64}-\frac {\left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {417}}{64}+\frac {i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}}{12}+\frac {61 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}}{192}+\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {1}{3} \\ -\frac {4 \left (2 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}+61 i \sqrt {3}+3 \sqrt {417}+9 i \sqrt {139}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+61\right )}{-\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+3 i \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+19 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-2 \sqrt {417}-6 i \sqrt {139}-46 i \sqrt {3}+2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-46} \\ 1 \end {array}\right ] \] Let $t = 1$ the eigenvector becomes \[ \left [\begin {array}{c} -\frac {\left (61 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}-9 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {139}+3 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {417}-16 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-61 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-16 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-64\right ) t}{192} \\ -\frac {4 t \left (2 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}+61 \,\operatorname {I} \sqrt {3}+3 \sqrt {417}+9 \,\operatorname {I} \sqrt {139}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+61\right )}{-\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+3 \,\operatorname {I} \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+19 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-2 \sqrt {417}-6 \,\operatorname {I} \sqrt {139}-46 \,\operatorname {I} \sqrt {3}+2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-46} \\ t \end {array}\right ] = \left [\begin {array}{c} -\frac {61 i \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}}{192}+\frac {3 i \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {139}}{64}-\frac {\left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {417}}{64}+\frac {i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}}{12}+\frac {61 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}}{192}+\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {1}{3} \\ -\frac {4 \left (2 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}+61 i \sqrt {3}+3 \sqrt {417}+9 i \sqrt {139}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+61\right )}{-\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+3 i \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+19 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-2 \sqrt {417}-6 i \sqrt {139}-46 i \sqrt {3}+2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-46} \\ 1 \end {array}\right ] \] Which is normalized to \[ \left [\begin {array}{c} -\frac {\left (61 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}-9 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {139}+3 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {417}-16 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-61 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-16 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-64\right ) t}{192} \\ -\frac {4 t \left (2 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}+61 \,\operatorname {I} \sqrt {3}+3 \sqrt {417}+9 \,\operatorname {I} \sqrt {139}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+61\right )}{-\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+3 \,\operatorname {I} \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+19 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-2 \sqrt {417}-6 \,\operatorname {I} \sqrt {139}-46 \,\operatorname {I} \sqrt {3}+2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-46} \\ t \end {array}\right ] = \left [\begin {array}{c} -\frac {61 i \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}}{192}+\frac {3 i \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {139}}{64}-\frac {\left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {417}}{64}+\frac {i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}}{12}+\frac {61 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}}{192}+\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {1}{3} \\ -\frac {4 \left (2 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}+61 i \sqrt {3}+3 \sqrt {417}+9 i \sqrt {139}+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+61\right )}{-\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+3 i \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+19 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-2 \sqrt {417}-6 i \sqrt {139}-46 i \sqrt {3}+2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-46} \\ 1 \end {array}\right ] \] Considering the eigenvalue $\lambda _{3} = -\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}$

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 & 0 \\ 1 & 0 & 2 \\ -1 & 0 & 1 \end {array}\right ] - \left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\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 {-i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}-8 i \sqrt {3}+\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-8}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}} & -1 & 0 \\ 1 & \frac {-i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}+\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}-8 i \sqrt {3}-8 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-8}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}} & 2 \\ -1 & 0 & \frac {-i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}-8 i \sqrt {3}+\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-8}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\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 \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {1}{3}+\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}&-1&0&0\\ 1&-\frac {2}{3}+\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}&2&0\\ -1&0&\frac {1}{3}+\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}-\frac {R_{1}}{\frac {1}{3}+\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\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 {-i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}-8 i \sqrt {3}+\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-8}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}&-1&0&0\\ 0&\frac {38 \left (\left (i+\frac {\sqrt {139}}{19}\right ) \sqrt {3}+\frac {3 i \sqrt {139}}{19}+1\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-92 i \sqrt {3}-12 i \sqrt {139}+4 \sqrt {3}\, \sqrt {139}-4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+92}{\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}} \left (i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}+8 i \sqrt {3}-\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}-4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}+8\right )}&2&0\\ -1&0&\frac {-i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}-8 i \sqrt {3}+\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-8}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}+\frac {12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}} R_{1}}{-i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}-8 i \sqrt {3}+\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-8} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {-i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}-8 i \sqrt {3}+\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-8}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}&-1&0&0\\ 0&\frac {38 \left (\left (i+\frac {\sqrt {139}}{19}\right ) \sqrt {3}+\frac {3 i \sqrt {139}}{19}+1\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-92 i \sqrt {3}-12 i \sqrt {139}+4 \sqrt {3}\, \sqrt {139}-4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+92}{\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}} \left (i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}+8 i \sqrt {3}-\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}-4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}+8\right )}&2&0\\ 0&\frac {12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}{i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}+8 i \sqrt {3}-\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}-4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}+8}&\frac {-i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}-8 i \sqrt {3}+\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-8}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}-\frac {6 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} R_{2}}{19 \left (\left (i+\frac {\sqrt {139}}{19}\right ) \sqrt {3}+\frac {3 i \sqrt {139}}{19}+1\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-46 i \sqrt {3}-6 i \sqrt {139}+2 \sqrt {3}\, \sqrt {139}-2 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+46} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {-i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}-8 i \sqrt {3}+\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-8}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}&-1&0&0\\ 0&\frac {38 \left (\left (i+\frac {\sqrt {139}}{19}\right ) \sqrt {3}+\frac {3 i \sqrt {139}}{19}+1\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-92 i \sqrt {3}-12 i \sqrt {139}+4 \sqrt {3}\, \sqrt {139}-4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+92}{\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}} \left (i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}+8 i \sqrt {3}-\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}-4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}+8\right )}&2&0\\ 0&0&0&0 \end {array} \right ] \end {align*}

Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} \frac {-i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}-8 i \sqrt {3}+\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-8}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}} & -1 & 0 \\ 0 & \frac {38 \left (\left (i+\frac {\sqrt {139}}{19}\right ) \sqrt {3}+\frac {3 i \sqrt {139}}{19}+1\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-92 i \sqrt {3}-12 i \sqrt {139}+4 \sqrt {3}\, \sqrt {139}-4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+92}{\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}} \left (i \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \sqrt {3}+8 i \sqrt {3}-\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}-4 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}+8\right )} & 2 \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] 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 \(\left \{v_{1} = -\frac {\left (2 i \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}+61 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-3 \sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+9 i \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-12 \sqrt {417}+2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-61 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-244\right ) t}{12 \left (61+3 \sqrt {417}\right )}, v_{2} = -\frac {4 t \left (2 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-3 \sqrt {417}+9 i \sqrt {139}+61 i \sqrt {3}-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-61\right )}{19 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}+3 i \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-46 i \sqrt {3}-6 i \sqrt {139}-2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \sqrt {417}+19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+46}\right \}\)

Hence the solution is \[ \left [\begin {array}{c} -\frac {\left (2 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}+61 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-3 \sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+9 \,\operatorname {I} \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-12 \sqrt {417}+2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-61 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-244\right ) t}{12 \left (61+3 \sqrt {417}\right )} \\ -\frac {4 t \left (2 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-3 \sqrt {417}+9 \,\operatorname {I} \sqrt {139}+61 \,\operatorname {I} \sqrt {3}-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-61\right )}{19 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}+3 \,\operatorname {I} \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-46 \,\operatorname {I} \sqrt {3}-6 \,\operatorname {I} \sqrt {139}-2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \sqrt {417}+19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+46} \\ t \end {array}\right ] = \left [\begin {array}{c} -\frac {\left (2 i \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}+61 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-3 \sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+9 i \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-12 \sqrt {417}+2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-61 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-244\right ) t}{12 \left (61+3 \sqrt {417}\right )} \\ -\frac {4 t \left (2 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-3 \sqrt {417}+9 i \sqrt {139}+61 i \sqrt {3}-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-61\right )}{19 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}+3 i \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-46 i \sqrt {3}-6 i \sqrt {139}-2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \sqrt {417}+19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+46} \\ 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 {\left (2 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}+61 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-3 \sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+9 \,\operatorname {I} \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-12 \sqrt {417}+2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-61 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-244\right ) t}{12 \left (61+3 \sqrt {417}\right )} \\ -\frac {4 t \left (2 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-3 \sqrt {417}+9 \,\operatorname {I} \sqrt {139}+61 \,\operatorname {I} \sqrt {3}-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-61\right )}{19 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}+3 \,\operatorname {I} \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-46 \,\operatorname {I} \sqrt {3}-6 \,\operatorname {I} \sqrt {139}-2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \sqrt {417}+19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+46} \\ t \end {array}\right ] = t \left [\begin {array}{c} -\frac {2 i \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}+61 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-3 \sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+9 i \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-12 \sqrt {417}+2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-61 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-244}{12 \left (61+3 \sqrt {417}\right )} \\ -\frac {4 \left (2 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-3 \sqrt {417}+9 i \sqrt {139}+61 i \sqrt {3}-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-61\right )}{19 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}+3 i \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-46 i \sqrt {3}-6 i \sqrt {139}-2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \sqrt {417}+19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+46} \\ 1 \end {array}\right ] \] Let $t = 1$ the eigenvector becomes \[ \left [\begin {array}{c} -\frac {\left (2 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}+61 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-3 \sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+9 \,\operatorname {I} \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-12 \sqrt {417}+2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-61 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-244\right ) t}{12 \left (61+3 \sqrt {417}\right )} \\ -\frac {4 t \left (2 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-3 \sqrt {417}+9 \,\operatorname {I} \sqrt {139}+61 \,\operatorname {I} \sqrt {3}-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-61\right )}{19 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}+3 \,\operatorname {I} \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-46 \,\operatorname {I} \sqrt {3}-6 \,\operatorname {I} \sqrt {139}-2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \sqrt {417}+19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+46} \\ t \end {array}\right ] = \left [\begin {array}{c} -\frac {2 i \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}+61 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-3 \sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+9 i \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-12 \sqrt {417}+2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-61 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-244}{12 \left (61+3 \sqrt {417}\right )} \\ -\frac {4 \left (2 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-3 \sqrt {417}+9 i \sqrt {139}+61 i \sqrt {3}-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-61\right )}{19 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}+3 i \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-46 i \sqrt {3}-6 i \sqrt {139}-2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \sqrt {417}+19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+46} \\ 1 \end {array}\right ] \] Which is normalized to \[ \left [\begin {array}{c} -\frac {\left (2 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}+61 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-3 \sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+9 \,\operatorname {I} \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-12 \sqrt {417}+2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-61 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-244\right ) t}{12 \left (61+3 \sqrt {417}\right )} \\ -\frac {4 t \left (2 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-3 \sqrt {417}+9 \,\operatorname {I} \sqrt {139}+61 \,\operatorname {I} \sqrt {3}-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-61\right )}{19 \,\operatorname {I} \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}+3 \,\operatorname {I} \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-46 \,\operatorname {I} \sqrt {3}-6 \,\operatorname {I} \sqrt {139}-2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \sqrt {417}+19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+46} \\ t \end {array}\right ] = \left [\begin {array}{c} -\frac {2 i \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}+61 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-3 \sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+9 i \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-12 \sqrt {417}+2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-61 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-244}{12 \left (61+3 \sqrt {417}\right )} \\ -\frac {4 \left (2 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-3 \sqrt {417}+9 i \sqrt {139}+61 i \sqrt {3}-\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-61\right )}{19 i \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}+3 i \sqrt {139}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+\sqrt {417}\, \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-46 i \sqrt {3}-6 i \sqrt {139}-2 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+2 \sqrt {417}+19 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+46} \\ 1 \end {array}\right ] \] 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

$\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}$	$1$	$1$	No	$\left [\begin {array}{c} -\frac {4}{\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )^{2}\right ) \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )} \\ \frac {4}{\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )^{2}} \\ 1 \end {array}\right ]$

$-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}$	$1$	$1$	No	\(\left [\begin {array}{c} -\frac {4}{\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {4}{-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\)

$-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}$	$1$	$1$	No	\(\left [\begin {array}{c} -\frac {4}{\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {4}{-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\)

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 $\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}$ is real and distinct then the corresponding eigenvector solution is \begin {align*} \vec {x}_{1}(t) &= \vec {v}_{1} e^{\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right ) t}\\ &= \left [\begin {array}{c} -\frac {4}{\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )^{2}\right ) \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )} \\ \frac {4}{\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )^{2}} \\ 1 \end {array}\right ] e^{\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{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 \\ z \left (t \right ) \end {array}\right ] &= c_{1} \left [\begin {array}{c} -\frac {4 \,{\mathrm e}^{\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right ) t}}{\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )^{2}\right ) \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )} \\ \frac {4 \,{\mathrm e}^{\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right ) t}}{\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )^{2}} \\ {\mathrm e}^{\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right ) t} \end {array}\right ] + c_{2} \left [\begin {array}{c} -\frac {4 \,{\mathrm e}^{\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}}{\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {4 \,{\mathrm e}^{\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}}{-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ {\mathrm e}^{\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} \end {array}\right ] + c_{3} \left [\begin {array}{c} -\frac {4 \,{\mathrm e}^{\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}}{\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {4 \,{\mathrm e}^{\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}}{-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ {\mathrm e}^{\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} \end {array}\right ] \end {align*}

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

9.4.3 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, y^{\prime }=x \left (t \right )+2 z \left (t \right ), z^{\prime }\left (t \right )=-x \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 \\ 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 & 0 \\ 1 & 0 & 2 \\ -1 & 0 & 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 & 0 \\ 1 & 0 & 2 \\ -1 & 0 & 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 & 0 \\ 1 & 0 & 2 \\ -1 & 0 & 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 (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}, \left [\begin {array}{c} -\frac {4}{\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )^{2}\right ) \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )} \\ \frac {4}{\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )^{2}} \\ 1 \end {array}\right ]\right ], \left [-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} -\frac {4}{\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {4}{-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\right ], \left [-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} -\frac {4}{\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {4}{-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}, \left [\begin {array}{c} -\frac {4}{\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )^{2}\right ) \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )} \\ \frac {4}{\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{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 (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right ) t}\cdot \left [\begin {array}{c} -\frac {4}{\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )^{2}\right ) \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )} \\ \frac {4}{\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )^{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} -\frac {4}{\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {4}{-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\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 (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}\cdot \left [\begin {array}{c} -\frac {4}{\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {4}{-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\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 (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right ) t}\cdot \left (\cos \left (\frac {\sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) t}{2}\right )\right )\cdot \left [\begin {array}{c} -\frac {4}{\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {4}{-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right ) t}\cdot \left [\begin {array}{c} -\frac {4 \left (\cos \left (\frac {\sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) t}{2}\right )\right )}{\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {4 \left (\cos \left (\frac {\sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) t}{2}\right )\right )}{-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \cos \left (\frac {\sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\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 (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right ) t}\cdot \left [\begin {array}{c} \frac {20736 \left (61+3 \sqrt {417}\right ) \left (\left (244+12 \sqrt {417}\right )^{\frac {5}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )+\left (244+12 \sqrt {417}\right )^{\frac {5}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-2 \left (244+12 \sqrt {417}\right )^{\frac {4}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )+2 \left (244+12 \sqrt {417}\right )^{\frac {4}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-16 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-16 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-64 \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-5856 \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )+64 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-288 \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \sqrt {417}\right )}{\left (\left (244+12 \sqrt {417}\right )^{\frac {8}{3}}-14 \left (244+12 \sqrt {417}\right )^{\frac {7}{3}}-1064 \left (244+12 \sqrt {417}\right )^{\frac {5}{3}}+5104 \left (244+12 \sqrt {417}\right )^{\frac {4}{3}}+8704 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+18344448+7168 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+898560 \sqrt {417}\right ) \left (\left (244+12 \sqrt {417}\right )^{\frac {4}{3}}+552+24 \sqrt {417}+12 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-16 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}\right )} \\ -\frac {72 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \left (\left (244+12 \sqrt {417}\right )^{\frac {4}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )+\left (244+12 \sqrt {417}\right )^{\frac {4}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-120 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-112 \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-168 \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {417}-112 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )+168 \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \sqrt {417}-3480 \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+3480 \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )\right )}{\left (244+12 \sqrt {417}\right )^{\frac {8}{3}}-14 \left (244+12 \sqrt {417}\right )^{\frac {7}{3}}-1064 \left (244+12 \sqrt {417}\right )^{\frac {5}{3}}+5104 \left (244+12 \sqrt {417}\right )^{\frac {4}{3}}+8704 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+18344448+7168 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+898560 \sqrt {417}} \\ \cos \left (\frac {\sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) t}{2}\right ) \end {array}\right ], {\moverset {\rightarrow }{x}}_{3}\left (t \right )={\mathrm e}^{\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right ) t}\cdot \left [\begin {array}{c} \frac {20736 \left (61+3 \sqrt {417}\right ) \left (\left (244+12 \sqrt {417}\right )^{\frac {5}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-\left (244+12 \sqrt {417}\right )^{\frac {5}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-2 \left (244+12 \sqrt {417}\right )^{\frac {4}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-2 \left (244+12 \sqrt {417}\right )^{\frac {4}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-16 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )+16 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-64 \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}+5856 \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-64 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )+288 \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \sqrt {417}\right )}{\left (\left (244+12 \sqrt {417}\right )^{\frac {8}{3}}-14 \left (244+12 \sqrt {417}\right )^{\frac {7}{3}}-1064 \left (244+12 \sqrt {417}\right )^{\frac {5}{3}}+5104 \left (244+12 \sqrt {417}\right )^{\frac {4}{3}}+8704 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+18344448+7168 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+898560 \sqrt {417}\right ) \left (\left (244+12 \sqrt {417}\right )^{\frac {4}{3}}+552+24 \sqrt {417}+12 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-16 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}\right )} \\ -\frac {72 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \left (\left (244+12 \sqrt {417}\right )^{\frac {4}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-\left (244+12 \sqrt {417}\right )^{\frac {4}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )+120 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-112 \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-168 \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {417}+112 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-168 \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \sqrt {417}-3480 \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \sqrt {3}-3480 \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )\right )}{\left (244+12 \sqrt {417}\right )^{\frac {8}{3}}-14 \left (244+12 \sqrt {417}\right )^{\frac {7}{3}}-1064 \left (244+12 \sqrt {417}\right )^{\frac {5}{3}}+5104 \left (244+12 \sqrt {417}\right )^{\frac {4}{3}}+8704 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+18344448+7168 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+898560 \sqrt {417}} \\ -\sin \left (\frac {\sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\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 (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right ) t}\cdot \left [\begin {array}{c} -\frac {4}{\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )^{2}\right ) \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )} \\ \frac {4}{\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )^{2}} \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right ) t}\cdot \left [\begin {array}{c} \frac {20736 \left (61+3 \sqrt {417}\right ) \left (\left (244+12 \sqrt {417}\right )^{\frac {5}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )+\left (244+12 \sqrt {417}\right )^{\frac {5}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-2 \left (244+12 \sqrt {417}\right )^{\frac {4}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )+2 \left (244+12 \sqrt {417}\right )^{\frac {4}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-16 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-16 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-64 \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-5856 \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )+64 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-288 \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \sqrt {417}\right )}{\left (\left (244+12 \sqrt {417}\right )^{\frac {8}{3}}-14 \left (244+12 \sqrt {417}\right )^{\frac {7}{3}}-1064 \left (244+12 \sqrt {417}\right )^{\frac {5}{3}}+5104 \left (244+12 \sqrt {417}\right )^{\frac {4}{3}}+8704 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+18344448+7168 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+898560 \sqrt {417}\right ) \left (\left (244+12 \sqrt {417}\right )^{\frac {4}{3}}+552+24 \sqrt {417}+12 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-16 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}\right )} \\ -\frac {72 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \left (\left (244+12 \sqrt {417}\right )^{\frac {4}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )+\left (244+12 \sqrt {417}\right )^{\frac {4}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-120 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-112 \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-168 \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {417}-112 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )+168 \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \sqrt {417}-3480 \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+3480 \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )\right )}{\left (244+12 \sqrt {417}\right )^{\frac {8}{3}}-14 \left (244+12 \sqrt {417}\right )^{\frac {7}{3}}-1064 \left (244+12 \sqrt {417}\right )^{\frac {5}{3}}+5104 \left (244+12 \sqrt {417}\right )^{\frac {4}{3}}+8704 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+18344448+7168 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+898560 \sqrt {417}} \\ \cos \left (\frac {\sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) t}{2}\right ) \end {array}\right ]+c_{3} {\mathrm e}^{\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right ) t}\cdot \left [\begin {array}{c} \frac {20736 \left (61+3 \sqrt {417}\right ) \left (\left (244+12 \sqrt {417}\right )^{\frac {5}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-\left (244+12 \sqrt {417}\right )^{\frac {5}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-2 \left (244+12 \sqrt {417}\right )^{\frac {4}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-2 \left (244+12 \sqrt {417}\right )^{\frac {4}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-16 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )+16 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-64 \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}+5856 \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-64 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )+288 \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \sqrt {417}\right )}{\left (\left (244+12 \sqrt {417}\right )^{\frac {8}{3}}-14 \left (244+12 \sqrt {417}\right )^{\frac {7}{3}}-1064 \left (244+12 \sqrt {417}\right )^{\frac {5}{3}}+5104 \left (244+12 \sqrt {417}\right )^{\frac {4}{3}}+8704 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+18344448+7168 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+898560 \sqrt {417}\right ) \left (\left (244+12 \sqrt {417}\right )^{\frac {4}{3}}+552+24 \sqrt {417}+12 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-16 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}\right )} \\ -\frac {72 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \left (\left (244+12 \sqrt {417}\right )^{\frac {4}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-\left (244+12 \sqrt {417}\right )^{\frac {4}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )+120 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-112 \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sqrt {3}-168 \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {417}+112 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )-168 \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \sqrt {417}-3480 \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right ) \sqrt {3}-3480 \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )\right )}{\left (244+12 \sqrt {417}\right )^{\frac {8}{3}}-14 \left (244+12 \sqrt {417}\right )^{\frac {7}{3}}-1064 \left (244+12 \sqrt {417}\right )^{\frac {5}{3}}+5104 \left (244+12 \sqrt {417}\right )^{\frac {4}{3}}+8704 \left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+18344448+7168 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}+898560 \sqrt {417}} \\ -\sin \left (\frac {\sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\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 \\ z \left (t \right ) \end {array}\right ]=\left [\begin {array}{c} \frac {\left (\left (\left (\left (11187 c_{2} \sqrt {3}+33561 c_{3} \right ) \sqrt {139}+228445 \sqrt {3}\, c_{3} +228445 c_{2} \right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}+\left (\left (-183 c_{2} \sqrt {3}+549 c_{3} \right ) \sqrt {139}+3737 \sqrt {3}\, c_{3} -3737 c_{2} \right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+44748 c_{2} \sqrt {3}\, \sqrt {139}+913780 c_{2} \right ) \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}\right )+228445 \left (\left (\left (-\frac {11187 \sqrt {3}\, c_{3}}{228445}+\frac {33561 c_{2}}{228445}\right ) \sqrt {139}+c_{2} \sqrt {3}-c_{3} \right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}+\left (\left (\frac {3 \sqrt {3}\, c_{3}}{3745}+\frac {9 c_{2}}{3745}\right ) \sqrt {139}+\frac {3737 c_{2} \sqrt {3}}{228445}+\frac {3737 c_{3}}{228445}\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}-\frac {44748 c_{3} \sqrt {3}\, \sqrt {139}}{228445}-4 c_{3} \right ) \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}\right )\right ) {\mathrm e}^{-\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-8 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}}-22374 c_{1} \left (\left (\sqrt {3}\, \sqrt {139}+\frac {228445}{11187}\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}+\left (-\frac {61 \sqrt {3}\, \sqrt {139}}{3729}-\frac {3737}{11187}\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}-2 \sqrt {3}\, \sqrt {139}-\frac {456890}{11187}\right ) {\mathrm e}^{\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-8\right ) t}{6 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}}}{134244 \sqrt {3}\, \sqrt {139}+2741340} \\ -\frac {\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \left (\left (\left (\frac {\left (\left (-c_{3} -\frac {c_{2} \sqrt {3}}{3}\right ) \sqrt {139}-\frac {19 c_{2}}{3}-\frac {19 \sqrt {3}\, c_{3}}{3}\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}}{2}+c_{2} \left (\frac {61}{3}+\sqrt {3}\, \sqrt {139}\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}+65 \left (-c_{3} +\frac {c_{2} \sqrt {3}}{3}\right ) \sqrt {139}+\frac {1327 c_{2}}{3}-\frac {1327 \sqrt {3}\, c_{3}}{3}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}\right )-\left (\frac {\left (\left (-\frac {\sqrt {3}\, c_{3}}{3}+c_{2} \right ) \sqrt {139}+\frac {19 c_{2} \sqrt {3}}{3}-\frac {19 c_{3}}{3}\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}}{2}+c_{3} \left (\frac {61}{3}+\sqrt {3}\, \sqrt {139}\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}+65 \left (\frac {\sqrt {3}\, c_{3}}{3}+c_{2} \right ) \sqrt {139}+\frac {1327 c_{2} \sqrt {3}}{3}+\frac {1327 c_{3}}{3}\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}\right )\right ) {\mathrm e}^{-\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-8 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}}+\left (-\frac {2654}{3}+\frac {\left (\sqrt {3}\, \sqrt {139}+19\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}}{3}+\left (\frac {61}{3}+\sqrt {3}\, \sqrt {139}\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-\frac {130 \sqrt {3}\, \sqrt {139}}{3}\right ) c_{1} {\mathrm e}^{\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-8\right ) t}{6 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}}\right )}{8 \left (183 \sqrt {3}\, \sqrt {139}+3737\right )} \\ c_{1} {\mathrm e}^{\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-8\right ) t}{6 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}}+\cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}\right ) {\mathrm e}^{-\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-8 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}} c_{2} -\sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}\right ) {\mathrm e}^{-\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-8 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}} c_{3} \end {array}\right ] \\ \bullet & {} & \textrm {Solution to the system of ODEs}\hspace {3pt} \\ {} & {} & \left \{x \left (t \right )=\frac {\left (\left (\left (\left (11187 c_{2} \sqrt {3}+33561 c_{3} \right ) \sqrt {139}+228445 \sqrt {3}\, c_{3} +228445 c_{2} \right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}+\left (\left (-183 c_{2} \sqrt {3}+549 c_{3} \right ) \sqrt {139}+3737 \sqrt {3}\, c_{3} -3737 c_{2} \right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+44748 c_{2} \sqrt {3}\, \sqrt {139}+913780 c_{2} \right ) \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}\right )+228445 \left (\left (\left (-\frac {11187 \sqrt {3}\, c_{3}}{228445}+\frac {33561 c_{2}}{228445}\right ) \sqrt {139}+c_{2} \sqrt {3}-c_{3} \right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}+\left (\left (\frac {3 \sqrt {3}\, c_{3}}{3745}+\frac {9 c_{2}}{3745}\right ) \sqrt {139}+\frac {3737 c_{2} \sqrt {3}}{228445}+\frac {3737 c_{3}}{228445}\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}-\frac {44748 c_{3} \sqrt {3}\, \sqrt {139}}{228445}-4 c_{3} \right ) \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}\right )\right ) {\mathrm e}^{-\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-8 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}}-22374 c_{1} \left (\left (\sqrt {3}\, \sqrt {139}+\frac {228445}{11187}\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}+\left (-\frac {61 \sqrt {3}\, \sqrt {139}}{3729}-\frac {3737}{11187}\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}-2 \sqrt {3}\, \sqrt {139}-\frac {456890}{11187}\right ) {\mathrm e}^{\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-8\right ) t}{6 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}}}{134244 \sqrt {3}\, \sqrt {139}+2741340}, y=-\frac {\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}} \left (\left (\left (\frac {\left (\left (-c_{3} -\frac {c_{2} \sqrt {3}}{3}\right ) \sqrt {139}-\frac {19 c_{2}}{3}-\frac {19 \sqrt {3}\, c_{3}}{3}\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}}{2}+c_{2} \left (\frac {61}{3}+\sqrt {3}\, \sqrt {139}\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}+65 \left (-c_{3} +\frac {c_{2} \sqrt {3}}{3}\right ) \sqrt {139}+\frac {1327 c_{2}}{3}-\frac {1327 \sqrt {3}\, c_{3}}{3}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}\right )-\left (\frac {\left (\left (-\frac {\sqrt {3}\, c_{3}}{3}+c_{2} \right ) \sqrt {139}+\frac {19 c_{2} \sqrt {3}}{3}-\frac {19 c_{3}}{3}\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}}{2}+c_{3} \left (\frac {61}{3}+\sqrt {3}\, \sqrt {139}\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}+65 \left (\frac {\sqrt {3}\, c_{3}}{3}+c_{2} \right ) \sqrt {139}+\frac {1327 c_{2} \sqrt {3}}{3}+\frac {1327 c_{3}}{3}\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}\right )\right ) {\mathrm e}^{-\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-8 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}}+\left (-\frac {2654}{3}+\frac {\left (\sqrt {3}\, \sqrt {139}+19\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}}{3}+\left (\frac {61}{3}+\sqrt {3}\, \sqrt {139}\right ) \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}-\frac {130 \sqrt {3}\, \sqrt {139}}{3}\right ) c_{1} {\mathrm e}^{\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-8\right ) t}{6 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}}\right )}{8 \left (183 \sqrt {3}\, \sqrt {139}+3737\right )}, z \left (t \right )=c_{1} {\mathrm e}^{\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-8\right ) t}{6 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}}+\cos \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}\right ) {\mathrm e}^{-\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-8 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}} c_{2} -\sin \left (\frac {\sqrt {3}\, \left (\left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (244+12 \sqrt {3}\, \sqrt {139}\right )^{\frac {1}{3}}}\right ) {\mathrm e}^{-\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-8 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-8\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}} c_{3} \right \} \end {array} \]

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-\frac {\left (-8+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-8 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}} \cos \left (\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t \sqrt {3}\, 2^{\frac {1}{3}}}{24 \left (61+3 \sqrt {417}\right )^{\frac {1}{3}}}\right ) c_{3} -{\mathrm e}^{-\frac {\left (-8+\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}-8 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}\right ) t}{12 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}} \sin \left (\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+8\right ) t \sqrt {3}\, 2^{\frac {1}{3}}}{24 \left (61+3 \sqrt {417}\right )^{\frac {1}{3}}}\right ) c_{2} +c_{1} {\mathrm e}^{\frac {\left (\left (244+12 \sqrt {417}\right )^{\frac {2}{3}}+4 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}-8\right ) t}{6 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}} \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align*}

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


	multiplicity

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

\(\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\)	\(1\)	\(1\)	No	\(\left [\begin {array}{c} -\frac {4}{\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )^{2}\right ) \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )} \\ \frac {4}{\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )^{2}} \\ 1 \end {array}\right ]\)

\(-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\)	\(1\)	\(1\)	No	\(\left [\begin {array}{c} -\frac {4}{\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {4}{-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\)

\(-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\)	\(1\)	\(1\)	No	\(\left [\begin {array}{c} -\frac {4}{\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {4}{-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {5}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{12}+\frac {2}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (244+12 \sqrt {417}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\)

9.4 problem 4

9.4.1 Solution using Matrix exponential method

9.4.2 Solution using explicit Eigenvalue and Eigenvector method

9.4.3 Maple step by step solution