problem 3

Internal problem ID [6713]
Internal ﬁle name [OUTPUT/5961_Sunday_June_05_2022_04_04_34_PM_15973523/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: 3.
ODE order: 1.
ODE degree: 1.

Solve \begin {align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )+4 y-9 z \left (t \right )\\ y^{\prime }&=6 x \left (t \right )-y\\ z^{\prime }\left (t \right )&=10 x \left (t \right )+4 y+3 z \left (t \right ) \end {align*}

9.3.1 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} -3 & 4 & -9 \\ 6 & -1 & 0 \\ 10 & 4 & 3 \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.

9.3.2 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} -3 & 4 & -9 \\ 6 & -1 & 0 \\ 10 & 4 & 3 \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} -3-\lambda & 4 & -9 \\ 6 & -1-\lambda & 0 \\ 10 & 4 & 3-\lambda \end {array}\right ]\right ) &= 0 \end {align*}

Which gives the characteristic equation \begin {align*} \lambda ^{3}+\lambda ^{2}+57 \lambda +369&=0 \end {align*}

The roots of the above are the eigenvalues. \begin {align*} \lambda _1 &= -\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\\ \lambda _2 &= \frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\\ \lambda _3 &= \frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}


eigenvalue	algebraic multiplicity	type of eigenvalue

$\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}$	$1$	complex eigenvalue

$\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}$	$1$	complex eigenvalue

$-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}$	$1$	real eigenvalue

Considering the eigenvalue $\lambda _{1} = -\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}$

We need to solve $A \vec {v} = \lambda \vec {v}$ or $(A-\lambda I) \vec {v} = \vec {0}$ which becomes \begin {align*} \left (\left [\begin {array}{ccc} -3 & 4 & -9 \\ 6 & -1 & 0 \\ 10 & 4 & 3 \end {array}\right ] - \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccc} \frac {\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}} & 4 & -9 \\ 6 & -\frac {2 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}}{2}+\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+85\right )}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}} & 0 \\ 10 & 4 & \frac {\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+10 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170}{3 \left (4726+306 \sqrt {291}\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 {30 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} R_{1}}{\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}&4&-9&0\\ 0&-\frac {102 \left (\left (\sqrt {291}+21\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-10 \sqrt {291}-\frac {30 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}}{17}-60\right )}{\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \left (-\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+170\right )}&\frac {162 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170}&0\\ 0&\frac {4 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-152 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-680}{\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170}&\frac {102 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+130 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+204 \sqrt {291}+1462 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+12784}{\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170\right )}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}+\frac {\left (4 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-152 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-680\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \left (-\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+170\right ) R_{2}}{102 \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170\right ) \left (\left (\sqrt {291}+21\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-10 \sqrt {291}-\frac {30 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}}{17}-60\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}&4&-9&0\\ 0&-\frac {102 \left (\left (\sqrt {291}+21\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-10 \sqrt {291}-\frac {30 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}}{17}-60\right )}{\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \left (-\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+170\right )}&\frac {162 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170}&0\\ 0&0&0&0 \end {array} \right ] \end {align*}

Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} \frac {\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}} & 4 & -9 \\ 0 & -\frac {102 \left (\left (\sqrt {291}+21\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-10 \sqrt {291}-\frac {30 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}}{17}-60\right )}{\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \left (-\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+170\right )} & \frac {162 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170} \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] 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 $\left \{v_{1} = \frac {27 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} t \left (17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-18 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020\right )}{\left (17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-30 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020\right ) \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170\right )}, v_{2} = -\frac {27 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} t}{17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-30 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020}\right \}$

Hence the solution is \[ \left [\begin {array}{c} \frac {27 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} t \left (17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-18 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020\right )}{\left (17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-30 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020\right ) \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170\right )} \\ -\frac {27 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} t}{17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-30 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {27 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} t \left (17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-18 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020\right )}{\left (17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-30 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020\right ) \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170\right )} \\ -\frac {27 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} t}{17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-30 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020} \\ t \end {array}\right ] \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \left [\begin {array}{c} \frac {27 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} t \left (17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-18 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020\right )}{\left (17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-30 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020\right ) \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170\right )} \\ -\frac {27 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} t}{17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-30 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020} \\ t \end {array}\right ] = t \left [\begin {array}{c} \frac {27 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \left (17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-18 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020\right )}{\left (17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-30 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020\right ) \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170\right )} \\ -\frac {27 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}}{17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-30 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020} \\ 1 \end {array}\right ] \] Let $t = 1$ the eigenvector becomes \[ \left [\begin {array}{c} \frac {27 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} t \left (17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-18 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020\right )}{\left (17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-30 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020\right ) \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170\right )} \\ -\frac {27 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} t}{17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-30 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {27 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \left (17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-18 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020\right )}{\left (17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-30 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020\right ) \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170\right )} \\ -\frac {27 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}}{17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-30 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020} \\ 1 \end {array}\right ] \] Which is normalized to \[ \left [\begin {array}{c} \frac {27 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} t \left (17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-18 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020\right )}{\left (17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-30 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020\right ) \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170\right )} \\ -\frac {27 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} t}{17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-30 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {27 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \left (17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-18 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020\right )}{\left (17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-30 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020\right ) \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-8 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170\right )} \\ -\frac {27 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}}{17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-30 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020} \\ 1 \end {array}\right ] \] Considering the eigenvalue $\lambda _{2} = \frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}$

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} -3 & 4 & -9 \\ 6 & -1 & 0 \\ 10 & 4 & 3 \end {array}\right ] - \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right ) \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccc} -\frac {\left (1+i \sqrt {3}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}+16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}-170}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}} & 4 & -9 \\ 6 & -\frac {\left (1+i \sqrt {3}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}+4 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}-170}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}} & 0 \\ 10 & 4 & -\frac {\left (1+i \sqrt {3}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}-20 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}-170}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\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 \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {8}{3}-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}+\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}&4&-9&0\\ 6&-\frac {2}{3}-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}+\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}&0&0\\ 10&4&\frac {10}{3}-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}+\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}-\frac {6 R_{1}}{-\frac {8}{3}-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}+\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {\left (1+i \sqrt {3}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}+16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}-170}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}&4&-9&0\\ 0&\frac {2142 \left (i \sqrt {3}+\frac {i \sqrt {97}}{7}-\frac {\sqrt {291}}{21}-1\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+6120 i \sqrt {3}+3060 i \sqrt {97}+1020 \sqrt {291}-360 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+6120}{\left (170-16 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-i \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) \sqrt {3}-\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}&-\frac {324 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{i \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {3}+\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170 i \sqrt {3}+16 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170}&0\\ 10&4&-\frac {\left (1+i \sqrt {3}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}-20 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}-170}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}+\frac {60 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}} R_{1}}{\left (1+i \sqrt {3}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}+16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}-170} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {\left (1+i \sqrt {3}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}+16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}-170}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}&4&-9&0\\ 0&\frac {2142 \left (i \sqrt {3}+\frac {i \sqrt {97}}{7}-\frac {\sqrt {291}}{21}-1\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+6120 i \sqrt {3}+3060 i \sqrt {97}+1020 \sqrt {291}-360 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+6120}{\left (170-16 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-i \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) \sqrt {3}-\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}&-\frac {324 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{i \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {3}+\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170 i \sqrt {3}+16 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170}&0\\ 0&\frac {\left (4 i \sqrt {3}+4\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+680 i \sqrt {3}+304 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}-680}{i \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}+\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}+16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}-170}&\frac {1462 \left (-i \sqrt {3}-\frac {9 i \sqrt {97}}{43}+\frac {3 \sqrt {291}}{43}+1\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+12784+12784 i \sqrt {3}+612 i \sqrt {97}+204 \sqrt {291}-260 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}}{\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \left (-170+16 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+i \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) \sqrt {3}+\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}\right )}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}-\frac {\left (\left (4 i \sqrt {3}+4\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+680 i \sqrt {3}+304 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}-680\right ) \left (170-16 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-i \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) \sqrt {3}-\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} R_{2}}{2142 \left (i \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}+\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}+16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}-170\right ) \left (\left (i \sqrt {3}+\frac {i \sqrt {97}}{7}-\frac {\sqrt {291}}{21}-1\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+\frac {20 i \sqrt {3}}{7}+\frac {10 i \sqrt {97}}{7}+\frac {10 \sqrt {291}}{21}-\frac {20 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}}{119}+\frac {20}{7}\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {\left (1+i \sqrt {3}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}+16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}-170}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}&4&-9&0\\ 0&\frac {2142 \left (i \sqrt {3}+\frac {i \sqrt {97}}{7}-\frac {\sqrt {291}}{21}-1\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+6120 i \sqrt {3}+3060 i \sqrt {97}+1020 \sqrt {291}-360 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+6120}{\left (170-16 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-i \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) \sqrt {3}-\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}&-\frac {324 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{i \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {3}+\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170 i \sqrt {3}+16 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170}&0\\ 0&0&0&0 \end {array} \right ] \end {align*}

Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} -\frac {\left (1+i \sqrt {3}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}+16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}-170}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}} & 4 & -9 \\ 0 & \frac {2142 \left (i \sqrt {3}+\frac {i \sqrt {97}}{7}-\frac {\sqrt {291}}{21}-1\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+6120 i \sqrt {3}+3060 i \sqrt {97}+1020 \sqrt {291}-360 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+6120}{\left (170-16 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-i \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) \sqrt {3}-\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}} & -\frac {324 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{i \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {3}+\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170 i \sqrt {3}+16 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170} \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] 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 \(\left \{v_{1} = -\frac {54 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} t \left (-51 i \sqrt {97}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-357 i \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {3}+17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-510 i \sqrt {97}-1020 i \sqrt {3}+36 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020\right )}{\left (-357 i \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {3}-51 i \sqrt {97}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+17 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+60 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-1020 i \sqrt {3}-510 i \sqrt {97}-170 \sqrt {291}+357 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1020\right ) \left (i \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {3}+\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170 i \sqrt {3}+16 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170\right )}, v_{2} = \frac {54 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} t}{-357 i \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}} \sqrt {3}-51 i \sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+17 \sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}} \sqrt {3}+60 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}-1020 i \sqrt {3}-510 i \sqrt {97}-170 \sqrt {3}\, \sqrt {97}+357 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}-1020}\right \}\)

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} -3 & 4 & -9 \\ 6 & -1 & 0 \\ 10 & 4 & 3 \end {array}\right ] - \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right ) \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccc} \frac {\left (i \sqrt {3}-1\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}-16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+170}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}} & 4 & -9 \\ 6 & \frac {\left (i \sqrt {3}-1\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}-4 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+170}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}} & 0 \\ 10 & 4 & \frac {\left (i \sqrt {3}-1\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}+20 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+170}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\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 \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {8}{3}-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}+\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}&4&-9&0\\ 6&-\frac {2}{3}-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}+\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}&0&0\\ 10&4&\frac {10}{3}-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}+\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}-\frac {6 R_{1}}{-\frac {8}{3}-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}+\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {\left (i \sqrt {3}-1\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}-16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+170}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}&4&-9&0\\ 0&\frac {2142 \left (i \sqrt {3}+\frac {i \sqrt {97}}{7}+\frac {\sqrt {291}}{21}+1\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+6120 i \sqrt {3}+3060 i \sqrt {97}-1020 \sqrt {291}+360 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-6120}{\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \left (-170+16 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-i \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) \sqrt {3}+\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}\right )}&\frac {324 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}{i \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}-\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}-16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+170}&0\\ 10&4&\frac {\left (i \sqrt {3}-1\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}+20 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+170}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}-\frac {60 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}} R_{1}}{\left (i \sqrt {3}-1\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}-16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+170} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {\left (i \sqrt {3}-1\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}-16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+170}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}&4&-9&0\\ 0&\frac {2142 \left (i \sqrt {3}+\frac {i \sqrt {97}}{7}+\frac {\sqrt {291}}{21}+1\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+6120 i \sqrt {3}+3060 i \sqrt {97}-1020 \sqrt {291}+360 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-6120}{\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \left (-170+16 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-i \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) \sqrt {3}+\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}\right )}&\frac {324 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}{i \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}-\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}-16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+170}&0\\ 0&\frac {\left (4 i \sqrt {3}-4\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+680 i \sqrt {3}-304 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+680}{i \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}-\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}-16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+170}&\frac {1462 \left (-i \sqrt {3}-\frac {9 i \sqrt {97}}{43}-\frac {3 \sqrt {291}}{43}-1\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-12784+12784 i \sqrt {3}+612 i \sqrt {97}-204 \sqrt {291}+260 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}}{\left (170-16 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+i \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) \sqrt {3}-\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}-\frac {\left (\left (4 i \sqrt {3}-4\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+680 i \sqrt {3}-304 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+680\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \left (-170+16 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-i \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) \sqrt {3}+\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}\right ) R_{2}}{2142 \left (i \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}-\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}-16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+170\right ) \left (\left (i \sqrt {3}+\frac {i \sqrt {97}}{7}+\frac {\sqrt {291}}{21}+1\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+\frac {20 i \sqrt {3}}{7}+\frac {10 i \sqrt {97}}{7}-\frac {10 \sqrt {291}}{21}+\frac {20 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}}{119}-\frac {20}{7}\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {\left (i \sqrt {3}-1\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}-16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+170}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}&4&-9&0\\ 0&\frac {2142 \left (i \sqrt {3}+\frac {i \sqrt {97}}{7}+\frac {\sqrt {291}}{21}+1\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+6120 i \sqrt {3}+3060 i \sqrt {97}-1020 \sqrt {291}+360 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-6120}{\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \left (-170+16 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-i \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) \sqrt {3}+\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}\right )}&\frac {324 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}{i \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}-\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}-16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+170}&0\\ 0&0&0&0 \end {array} \right ] \end {align*}

Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} \frac {\left (i \sqrt {3}-1\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}-16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+170}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}} & 4 & -9 \\ 0 & \frac {2142 \left (i \sqrt {3}+\frac {i \sqrt {97}}{7}+\frac {\sqrt {291}}{21}+1\right ) \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+6120 i \sqrt {3}+3060 i \sqrt {97}-1020 \sqrt {291}+360 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-6120}{\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \left (-170+16 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-i \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) \sqrt {3}+\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}\right )} & \frac {324 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}{i \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}-\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}-16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+170} \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] 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 {18 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}} t \left (9 i \sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+3 \sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}+8 i \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}-207 i \sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+69 \sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}} \sqrt {3}-1349 i \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}} \sqrt {3}+5148 \sqrt {3}\, \sqrt {97}+8 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+1349 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+79508\right )}{\left (-4 i \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}+3 i \sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+\sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}-635 i \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}} \sqrt {3}-105 i \sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+35 \sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}} \sqrt {3}-4 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+1908 \sqrt {3}\, \sqrt {97}+635 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+29468\right ) \left (i \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}-\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}-16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+170\right )}, v_{2} = -\frac {3 t \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \left (85 i \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {3}+2363 i \sqrt {3}-153 \sqrt {291}+459 i \sqrt {97}-8 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+85 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-2363\right )}{17 \left (-4 i \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {3}+3 i \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {97}+\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {291}-635 i \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {3}-105 i \sqrt {97}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+35 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-4 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+1908 \sqrt {291}+635 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+29468\right )}\right \}\)

Hence the solution is \[ \text {Expression too large to display} \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \left [\begin {array}{c} \frac {18 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}} t \left (9 \,\operatorname {I} \sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+3 \sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}+8 \,\operatorname {I} \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}-207 \,\operatorname {I} \sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+69 \sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}} \sqrt {3}-1349 \,\operatorname {I} \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}} \sqrt {3}+5148 \sqrt {3}\, \sqrt {97}+8 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+1349 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+79508\right )}{\left (-4 \,\operatorname {I} \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}+3 \,\operatorname {I} \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {97}+\sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}-635 \,\operatorname {I} \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}} \sqrt {3}-105 \,\operatorname {I} \sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+35 \sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}} \sqrt {3}-4 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+1908 \sqrt {3}\, \sqrt {97}+635 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+29468\right ) \left (\operatorname {I} \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}-\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 \,\operatorname {I} \sqrt {3}-16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+170\right )} \\ -\frac {3 t \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \left (85 \,\operatorname {I} \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {3}+2363 \,\operatorname {I} \sqrt {3}-153 \sqrt {291}+459 \,\operatorname {I} \sqrt {97}-8 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+85 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-2363\right )}{17 \left (-4 \,\operatorname {I} \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {3}+3 \,\operatorname {I} \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {97}+\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {291}-635 \,\operatorname {I} \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {3}-105 \,\operatorname {I} \sqrt {97}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+35 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-4 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+1908 \sqrt {291}+635 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+29468\right )} \\ t \end {array}\right ] = t \left [\begin {array}{c} \frac {18 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}} \left (9 i \sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+3 \sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}+8 i \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}-207 i \sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+69 \sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}} \sqrt {3}-1349 i \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}} \sqrt {3}+5148 \sqrt {3}\, \sqrt {97}+8 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+1349 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+79508\right )}{\left (-4 i \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}+3 i \sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+\sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}-635 i \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}} \sqrt {3}-105 i \sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+35 \sqrt {97}\, \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}} \sqrt {3}-4 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+1908 \sqrt {3}\, \sqrt {97}+635 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+29468\right ) \left (i \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \sqrt {3}-\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170 i \sqrt {3}-16 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+170\right )} \\ -\frac {3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \left (85 i \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {3}+2363 i \sqrt {3}-153 \sqrt {291}+459 i \sqrt {97}-8 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+85 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-2363\right )}{17 \left (-4 i \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {3}+3 i \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {97}+\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {291}-635 i \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {3}-105 i \sqrt {97}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+35 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-4 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+1908 \sqrt {291}+635 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+29468\right )} \\ 1 \end {array}\right ] \] Let $t = 1$ the eigenvector becomes \[ \text {Expression too large to display} \] Which is normalized to \[ \text {Expression too large to display} \] The following table gives a summary of this result. It shows for each eigenvalue the algebraic multiplicity $m$, and its geometric multiplicity $k$ and the eigenvectors associated with the eigenvalue. If $m>k$ then the eigenvalue is defective which means the number of normal linearly independent eigenvectors associated with this eigenvalue (called the geometric multiplicity $k$) does not equal the algebraic multiplicity $m$, and we need to determine an additional $m-k$ generalized eigenvectors for this eigenvalue.


	multiplicity

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

$-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}$	$1$	$1$	No	\(\left [\begin {array}{c} -\frac {135 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )}{\left (-\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {1870}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}+2 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}\right ) \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {8}{3}\right )} \\ \frac {\frac {9 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}}{2}+3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-\frac {510}{\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {573}{2}}{-\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {1870}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}+2 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}} \\ 1 \end {array}\right ]\)

$\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}$	$1$	$1$	No	\(\left [\begin {array}{c} -\frac {135 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {935}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}+\frac {11 i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}+2 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {8}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {\frac {9 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}}{2}-\frac {3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{2}+\frac {255}{\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {573}{2}-\frac {9 i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}}{\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {935}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}+\frac {11 i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}+2 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\)

$\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}$	$1$	$1$	No	\(\left [\begin {array}{c} -\frac {135 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {935}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}-\frac {11 i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}+2 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {8}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {\frac {9 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}}{2}-\frac {3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{2}+\frac {255}{\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {573}{2}+\frac {9 i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}}{\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {935}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}-\frac {11 i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}+2 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\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 $-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}$ is real and distinct then the corresponding eigenvector solution is \begin {align*} \vec {x}_{1}(t) &= \vec {v}_{1} e^{\left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t}\\ &= \left [\begin {array}{c} -\frac {135 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )}{\left (-\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {1870}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}+2 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}\right ) \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {8}{3}\right )} \\ \frac {\frac {9 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}}{2}+3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-\frac {510}{\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {573}{2}}{-\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {1870}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}+2 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}} \\ 1 \end {array}\right ] e^{\left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t} \end {align*}

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

Which is written as \begin {align*} \left [\begin {array}{c} x \left (t \right ) \\ y \\ z \left (t \right ) \end {array}\right ] &= c_{1} \left [\begin {array}{c} -\frac {135 \,{\mathrm e}^{\left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t} \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )}{\left (-\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {1870}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}+2 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}\right ) \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {8}{3}\right )} \\ \frac {9 \,{\mathrm e}^{\left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t} \left (\left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}+\frac {2 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {340}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {191}{3}\right )}{2 \left (-\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {1870}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}+2 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}\right )} \\ {\mathrm e}^{\left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t} \end {array}\right ] + c_{2} \text {Expression too large to display} + c_{3} \text {Expression too large to display} \end {align*}

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

9.3.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [x^{\prime }\left (t \right )=-3 x \left (t \right )+4 y-9 z \left (t \right ), y^{\prime }=6 x \left (t \right )-y, z^{\prime }\left (t \right )=10 x \left (t \right )+4 y+3 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} -3 & 4 & -9 \\ 6 & -1 & 0 \\ 10 & 4 & 3 \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} -3 & 4 & -9 \\ 6 & -1 & 0 \\ 10 & 4 & 3 \end {array}\right ]\cdot {\moverset {\rightarrow }{x}}\left (t \right ) \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{ccc} -3 & 4 & -9 \\ 6 & -1 & 0 \\ 10 & 4 & 3 \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 (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}, \left [\begin {array}{c} -\frac {135 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )}{\left (-\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {1870}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}+2 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}\right ) \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {8}{3}\right )} \\ \frac {9 \left (\left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}+\frac {2 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {340}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {191}{3}\right )}{2 \left (-\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {1870}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}+2 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}\right )} \\ 1 \end {array}\right ]\right ], \left [\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} -\frac {135 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {935}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}-\frac {11 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}+2 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {8}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {9 \left (\left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {191}{3}+\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )\right )}{2 \left (\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {935}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}-\frac {11 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}+2 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right )} \\ 1 \end {array}\right ]\right ], \left [\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} -\frac {135 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {935}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}+\frac {11 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}+2 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {8}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {9 \left (\left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {191}{3}-\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )\right )}{2 \left (\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {935}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}+\frac {11 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}+2 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right )} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}, \left [\begin {array}{c} -\frac {135 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )}{\left (-\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {1870}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}+2 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}\right ) \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {8}{3}\right )} \\ \frac {9 \left (\left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}+\frac {2 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {340}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {191}{3}\right )}{2 \left (-\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {1870}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}+2 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}\right )} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}_{1}={\mathrm e}^{\left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t}\cdot \left [\begin {array}{c} -\frac {135 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )}{\left (-\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {1870}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}+2 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}\right ) \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {8}{3}\right )} \\ \frac {9 \left (\left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}+\frac {2 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {340}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {191}{3}\right )}{2 \left (-\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {1870}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}+2 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}\right )} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} -\frac {135 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {935}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}-\frac {11 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}+2 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {8}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {9 \left (\left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {191}{3}+\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )\right )}{2 \left (\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {935}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}-\frac {11 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}+2 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right )} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}\cdot \left [\begin {array}{c} -\frac {135 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {935}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}-\frac {11 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}+2 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {8}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {9 \left (\left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {191}{3}+\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )\right )}{2 \left (\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {935}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}-\frac {11 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}+2 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right )} \\ 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 (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t}\cdot \left (\cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right ) t}{2}\right )\right )\cdot \left [\begin {array}{c} -\frac {135 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {935}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}-\frac {11 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}+2 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {8}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {9 \left (\left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {191}{3}+\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )\right )}{2 \left (\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {935}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}-\frac {11 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}+2 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right )} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t}\cdot \left [\begin {array}{c} -\frac {135 \left (\cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right ) t}{2}\right )\right ) \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {935}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}-\frac {11 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}+2 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {8}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {9 \left (\cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right ) t}{2}\right )\right ) \left (\left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {191}{3}+\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )\right )}{2 \left (\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {935}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}-\frac {11 \,\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}+2 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right )} \\ \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\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 (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t}\cdot \left [\begin {array}{c} -\frac {1215 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \left (2 \left (4726+306 \sqrt {291}\right )^{\frac {8}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-2 \left (4726+306 \sqrt {291}\right )^{\frac {8}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+45 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+41 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+6296 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+11548 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-1224 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {291}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-2040 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+109112 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-227664 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {291}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-12427000 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+624240 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {291}\, \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+24708667680 \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+29718856 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+1040400 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {291}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+230726040 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+46670055120 \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+1571212080 \sqrt {291}\, \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-165712600 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+2764134720 \sqrt {291}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )\right )}{4 \left (2 \left (4726+306 \sqrt {291}\right )^{\frac {8}{3}}+27 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}}+19067 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}}+612 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {291}+173889 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}}-341496 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {291}-9247966 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+1508580 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-19569455820-119177820 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1331503920 \sqrt {291}\right ) \left (\left (4726+306 \sqrt {291}\right )^{\frac {4}{3}}+234 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-1360 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+2448 \sqrt {291}+66708\right )} \\ \frac {9 \left (4 \left (4726+306 \sqrt {291}\right )^{\frac {8}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-47 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+11 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+11250 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+612 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {291}\, \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+7678 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+612 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {291}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+12172 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+78836 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-286416 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {291}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+2601000 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-832320 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {291}\, \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+4575991320 \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-47484536 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-832320 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {291}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-233939720 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-75954714120 \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+324916920 \sqrt {291}\, \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-76723720 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-4660263720 \sqrt {291}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )\right )}{8 \left (2 \left (4726+306 \sqrt {291}\right )^{\frac {8}{3}}+27 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}}+19067 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}}+612 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {291}+173889 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}}-341496 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {291}-9247966 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+1508580 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-19569455820-119177820 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1331503920 \sqrt {291}\right )} \\ \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right ) t}{2}\right ) \end {array}\right ], {\moverset {\rightarrow }{x}}_{3}\left (t \right )={\mathrm e}^{\left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t}\cdot \left [\begin {array}{c} \frac {1215 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \left (2 \left (4726+306 \sqrt {291}\right )^{\frac {8}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+2 \left (4726+306 \sqrt {291}\right )^{\frac {8}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+45 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-41 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+6296 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-11548 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+1224 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {291}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-2040 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-109112 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+227664 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {291}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-12427000 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+624240 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {291}\, \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+24708667680 \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-29718856 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-1040400 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {291}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+230726040 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-46670055120 \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+1571212080 \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {291}+165712600 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-2764134720 \sqrt {291}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )\right )}{4 \left (2 \left (4726+306 \sqrt {291}\right )^{\frac {8}{3}}+27 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}}+19067 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}}+612 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {291}+173889 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}}-341496 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {291}-9247966 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+1508580 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-19569455820-119177820 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1331503920 \sqrt {291}\right ) \left (\left (4726+306 \sqrt {291}\right )^{\frac {4}{3}}+234 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-1360 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+2448 \sqrt {291}+66708\right )} \\ \frac {9 \left (4 \left (4726+306 \sqrt {291}\right )^{\frac {8}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+47 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+11 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-11250 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-612 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {291}\, \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+7678 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+612 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {291}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-12172 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+78836 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-286416 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {291}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-2601000 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+832320 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {291}\, \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-4575991320 \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-47484536 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-832320 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {291}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+233939720 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-75954714120 \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-324916920 \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {291}-76723720 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-4660263720 \sqrt {291}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )\right )}{8 \left (2 \left (4726+306 \sqrt {291}\right )^{\frac {8}{3}}+27 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}}+19067 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}}+612 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {291}+173889 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}}-341496 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {291}-9247966 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+1508580 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-19569455820-119177820 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1331503920 \sqrt {291}\right )} \\ -\sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\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 (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t}\cdot \left [\begin {array}{c} -\frac {135 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )}{\left (-\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {1870}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}+2 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}\right ) \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {8}{3}\right )} \\ \frac {9 \left (\left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}+\frac {2 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {340}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {191}{3}\right )}{2 \left (-\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {1870}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}+2 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}\right )} \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{\left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t}\cdot \left [\begin {array}{c} -\frac {1215 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \left (2 \left (4726+306 \sqrt {291}\right )^{\frac {8}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-2 \left (4726+306 \sqrt {291}\right )^{\frac {8}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+45 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+41 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+6296 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+11548 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-1224 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {291}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-2040 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+109112 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-227664 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {291}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-12427000 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+624240 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {291}\, \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+24708667680 \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+29718856 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+1040400 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {291}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+230726040 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+46670055120 \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+1571212080 \sqrt {291}\, \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-165712600 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+2764134720 \sqrt {291}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )\right )}{4 \left (2 \left (4726+306 \sqrt {291}\right )^{\frac {8}{3}}+27 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}}+19067 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}}+612 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {291}+173889 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}}-341496 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {291}-9247966 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+1508580 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-19569455820-119177820 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1331503920 \sqrt {291}\right ) \left (\left (4726+306 \sqrt {291}\right )^{\frac {4}{3}}+234 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-1360 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+2448 \sqrt {291}+66708\right )} \\ \frac {9 \left (4 \left (4726+306 \sqrt {291}\right )^{\frac {8}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-47 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+11 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+11250 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+612 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {291}\, \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+7678 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+612 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {291}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+12172 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+78836 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-286416 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {291}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+2601000 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-832320 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {291}\, \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+4575991320 \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-47484536 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-832320 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {291}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-233939720 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-75954714120 \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+324916920 \sqrt {291}\, \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-76723720 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-4660263720 \sqrt {291}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )\right )}{8 \left (2 \left (4726+306 \sqrt {291}\right )^{\frac {8}{3}}+27 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}}+19067 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}}+612 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {291}+173889 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}}-341496 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {291}-9247966 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+1508580 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-19569455820-119177820 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1331503920 \sqrt {291}\right )} \\ \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right ) t}{2}\right ) \end {array}\right ]+c_{3} {\mathrm e}^{\left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right ) t}\cdot \left [\begin {array}{c} \frac {1215 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \left (2 \left (4726+306 \sqrt {291}\right )^{\frac {8}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+2 \left (4726+306 \sqrt {291}\right )^{\frac {8}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+45 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-41 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+6296 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-11548 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+1224 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {291}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-2040 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-109112 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+227664 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {291}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-12427000 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+624240 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {291}\, \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+24708667680 \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-29718856 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-1040400 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {291}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+230726040 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-46670055120 \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+1571212080 \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {291}+165712600 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-2764134720 \sqrt {291}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )\right )}{4 \left (2 \left (4726+306 \sqrt {291}\right )^{\frac {8}{3}}+27 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}}+19067 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}}+612 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {291}+173889 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}}-341496 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {291}-9247966 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+1508580 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-19569455820-119177820 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1331503920 \sqrt {291}\right ) \left (\left (4726+306 \sqrt {291}\right )^{\frac {4}{3}}+234 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-1360 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+2448 \sqrt {291}+66708\right )} \\ \frac {9 \left (4 \left (4726+306 \sqrt {291}\right )^{\frac {8}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+47 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+11 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-11250 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-612 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {291}\, \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+7678 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+612 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {291}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-12172 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+78836 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-286416 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {291}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-2601000 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+832320 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {291}\, \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-4575991320 \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-47484536 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-832320 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {291}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )+233939720 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-75954714120 \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-324916920 \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {291}-76723720 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )-4660263720 \sqrt {291}\, \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )\right )}{8 \left (2 \left (4726+306 \sqrt {291}\right )^{\frac {8}{3}}+27 \left (4726+306 \sqrt {291}\right )^{\frac {7}{3}}+19067 \left (4726+306 \sqrt {291}\right )^{\frac {5}{3}}+612 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}} \sqrt {291}+173889 \left (4726+306 \sqrt {291}\right )^{\frac {4}{3}}-341496 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}} \sqrt {291}-9247966 \left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+1508580 \sqrt {291}\, \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-19569455820-119177820 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-1331503920 \sqrt {291}\right )} \\ -\sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\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 {625500 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \left (\left (\left (\left (\left (-\frac {5707 c_{3}}{69500}-\frac {5707 c_{2} \sqrt {3}}{208500}\right ) \sqrt {97}-\frac {291341 \sqrt {3}\, c_{3}}{625500}-\frac {291341 c_{2}}{625500}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+c_{2} \left (\frac {21446}{1251}+\sqrt {3}\, \sqrt {97}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+\left (-\frac {244069 c_{2} \sqrt {3}}{104250}+\frac {244069 c_{3}}{34750}\right ) \sqrt {97}-\frac {12368197 c_{2}}{312750}+\frac {12368197 \sqrt {3}\, c_{3}}{312750}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}\right )+\sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}\right ) \left (\left (\left (\frac {5707 c_{2}}{69500}-\frac {5707 \sqrt {3}\, c_{3}}{208500}\right ) \sqrt {97}+\frac {291341 c_{2} \sqrt {3}}{625500}-\frac {291341 c_{3}}{625500}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+c_{3} \left (\frac {21446}{1251}+\sqrt {3}\, \sqrt {97}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+\left (-\frac {244069 \sqrt {3}\, c_{3}}{104250}-\frac {244069 c_{2}}{34750}\right ) \sqrt {97}-\frac {12368197 c_{3}}{312750}-\frac {12368197 c_{2} \sqrt {3}}{312750}\right )\right ) {\mathrm e}^{-\frac {\left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}}{2}+\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+85\right ) t}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}}+{\mathrm e}^{-\frac {\left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170\right ) t}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}} \left (\left (\frac {5707 \sqrt {3}\, \sqrt {97}}{104250}+\frac {291341}{312750}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+\left (\frac {21446}{1251}+\sqrt {3}\, \sqrt {97}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+\frac {12368197}{156375}+\frac {244069 \sqrt {3}\, \sqrt {97}}{52125}\right ) c_{1} \right )}{7185428352 \sqrt {3}\, \sqrt {97}+122544196992} \\ \frac {\left (\left (\left (\left (-25557 c_{2} \sqrt {3}-76671 c_{3} \right ) \sqrt {97}-427297 \sqrt {3}\, c_{3} -427297 c_{2} \right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+\left (\left (-1740 c_{2} \sqrt {3}+5220 c_{3} \right ) \sqrt {97}+30415 \sqrt {3}\, c_{3} -30415 c_{2} \right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+1926540 c_{2} \sqrt {3}\, \sqrt {97}+33026840 c_{2} \right ) \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}\right )+427297 \left (\left (\left (-\frac {25557 \sqrt {3}\, c_{3}}{427297}+\frac {76671 c_{2}}{427297}\right ) \sqrt {97}+c_{2} \sqrt {3}-c_{3} \right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+\left (\left (-\frac {1740 \sqrt {3}\, c_{3}}{427297}-\frac {5220 c_{2}}{427297}\right ) \sqrt {97}-\frac {30415 c_{2} \sqrt {3}}{427297}-\frac {30415 c_{3}}{427297}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+\frac {1926540 c_{3} \left (\frac {21446}{1251}+\sqrt {3}\, \sqrt {97}\right )}{427297}\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}\right )\right ) {\mathrm e}^{-\frac {\left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}}{2}+\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+85\right ) t}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}}+51114 \,{\mathrm e}^{-\frac {\left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170\right ) t}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}} c_{1} \left (\left (\sqrt {3}\, \sqrt {97}+\frac {427297}{25557}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+\left (\frac {580 \sqrt {3}\, \sqrt {97}}{8519}+\frac {4345}{3651}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+\frac {45870 \sqrt {3}\, \sqrt {97}}{1217}+\frac {2359060}{3651}\right )}{1441152 \sqrt {3}\, \sqrt {97}+24705792} \\ {\mathrm e}^{-\frac {\left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}}{2}+\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+85\right ) t}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}\right ) c_{2} +{\mathrm e}^{-\frac {\left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}}{2}+\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+85\right ) t}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}\right ) c_{3} +{\mathrm e}^{-\frac {\left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170\right ) t}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}} c_{1} \end {array}\right ] \\ \bullet & {} & \textrm {Solution to the system of ODEs}\hspace {3pt} \\ {} & {} & \left \{x \left (t \right )=-\frac {625500 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}} \left (\left (\left (\left (\left (-\frac {5707 c_{3}}{69500}-\frac {5707 c_{2} \sqrt {3}}{208500}\right ) \sqrt {97}-\frac {291341 \sqrt {3}\, c_{3}}{625500}-\frac {291341 c_{2}}{625500}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+c_{2} \left (\frac {21446}{1251}+\sqrt {3}\, \sqrt {97}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+\left (-\frac {244069 c_{2} \sqrt {3}}{104250}+\frac {244069 c_{3}}{34750}\right ) \sqrt {97}-\frac {12368197 c_{2}}{312750}+\frac {12368197 \sqrt {3}\, c_{3}}{312750}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}\right )+\sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}\right ) \left (\left (\left (\frac {5707 c_{2}}{69500}-\frac {5707 \sqrt {3}\, c_{3}}{208500}\right ) \sqrt {97}+\frac {291341 c_{2} \sqrt {3}}{625500}-\frac {291341 c_{3}}{625500}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+c_{3} \left (\frac {21446}{1251}+\sqrt {3}\, \sqrt {97}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+\left (-\frac {244069 \sqrt {3}\, c_{3}}{104250}-\frac {244069 c_{2}}{34750}\right ) \sqrt {97}-\frac {12368197 c_{3}}{312750}-\frac {12368197 c_{2} \sqrt {3}}{312750}\right )\right ) {\mathrm e}^{-\frac {\left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}}{2}+\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+85\right ) t}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}}+{\mathrm e}^{-\frac {\left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170\right ) t}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}} \left (\left (\frac {5707 \sqrt {3}\, \sqrt {97}}{104250}+\frac {291341}{312750}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+\left (\frac {21446}{1251}+\sqrt {3}\, \sqrt {97}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+\frac {12368197}{156375}+\frac {244069 \sqrt {3}\, \sqrt {97}}{52125}\right ) c_{1} \right )}{7185428352 \sqrt {3}\, \sqrt {97}+122544196992}, y=\frac {\left (\left (\left (\left (-25557 c_{2} \sqrt {3}-76671 c_{3} \right ) \sqrt {97}-427297 \sqrt {3}\, c_{3} -427297 c_{2} \right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+\left (\left (-1740 c_{2} \sqrt {3}+5220 c_{3} \right ) \sqrt {97}+30415 \sqrt {3}\, c_{3} -30415 c_{2} \right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+1926540 c_{2} \sqrt {3}\, \sqrt {97}+33026840 c_{2} \right ) \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}\right )+427297 \left (\left (\left (-\frac {25557 \sqrt {3}\, c_{3}}{427297}+\frac {76671 c_{2}}{427297}\right ) \sqrt {97}+c_{2} \sqrt {3}-c_{3} \right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+\left (\left (-\frac {1740 \sqrt {3}\, c_{3}}{427297}-\frac {5220 c_{2}}{427297}\right ) \sqrt {97}-\frac {30415 c_{2} \sqrt {3}}{427297}-\frac {30415 c_{3}}{427297}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+\frac {1926540 c_{3} \left (\frac {21446}{1251}+\sqrt {3}\, \sqrt {97}\right )}{427297}\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}\right )\right ) {\mathrm e}^{-\frac {\left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}}{2}+\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+85\right ) t}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}}+51114 \,{\mathrm e}^{-\frac {\left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170\right ) t}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}} c_{1} \left (\left (\sqrt {3}\, \sqrt {97}+\frac {427297}{25557}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}+\left (\frac {580 \sqrt {3}\, \sqrt {97}}{8519}+\frac {4345}{3651}\right ) \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+\frac {45870 \sqrt {3}\, \sqrt {97}}{1217}+\frac {2359060}{3651}\right )}{1441152 \sqrt {3}\, \sqrt {97}+24705792}, z \left (t \right )={\mathrm e}^{-\frac {\left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}}{2}+\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+85\right ) t}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}\right ) c_{2} +{\mathrm e}^{-\frac {\left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}}{2}+\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}+85\right ) t}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {2}{3}}+170\right ) t}{6 \left (4726+306 \sqrt {3}\, \sqrt {97}\right )^{\frac {1}{3}}}\right ) c_{3} +{\mathrm e}^{-\frac {\left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170\right ) t}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}} c_{1} \right \} \end {array} \]

\begin{align*} \text {Expression too large to display} \\ y \left (t \right ) &= \cos \left (\frac {\left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t \sqrt {3}\, 1156^{\frac {1}{3}}}{204 \left (139+9 \sqrt {291}\right )^{\frac {1}{3}}}\right ) {\mathrm e}^{\frac {\left (-170+\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-2 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}} c_{3} +{\mathrm e}^{\frac {\left (-170+\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}-2 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}\right ) t}{6 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}} \sin \left (\frac {\left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+170\right ) t \sqrt {3}\, 1156^{\frac {1}{3}}}{204 \left (139+9 \sqrt {291}\right )^{\frac {1}{3}}}\right ) c_{2} +c_{1} {\mathrm e}^{-\frac {\left (\left (4726+306 \sqrt {291}\right )^{\frac {2}{3}}+\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-170\right ) t}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}} \\ \text {Expression too large to display} \\ \end{align*}

\begin{align*} x(t)\to 4 c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2+57 \text {$\#$1}+369\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-12 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}+57}\&\right ]-9 c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2+57 \text {$\#$1}+369\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}+e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}+57}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2+57 \text {$\#$1}+369\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-2 \text {$\#$1} e^{\text {$\#$1} t}-3 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}+57}\&\right ] \\ y(t)\to -54 c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2+57 \text {$\#$1}+369\&,\frac {e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}+57}\&\right ]+6 c_1 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2+57 \text {$\#$1}+369\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-3 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}+57}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2+57 \text {$\#$1}+369\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+81 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}+57}\&\right ] \\ z(t)\to 4 c_2 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2+57 \text {$\#$1}+369\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}+13 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}+57}\&\right ]+2 c_1 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2+57 \text {$\#$1}+369\&,\frac {5 \text {$\#$1} e^{\text {$\#$1} t}+17 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}+57}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3+\text {$\#$1}^2+57 \text {$\#$1}+369\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+4 \text {$\#$1} e^{\text {$\#$1} t}-21 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2+2 \text {$\#$1}+57}\&\right ] \\ \end{align*}


	multiplicity

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

\(-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\)	\(1\)	\(1\)	No	\(\left [\begin {array}{c} -\frac {135 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )}{\left (-\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {1870}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}+2 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}\right ) \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {8}{3}\right )} \\ \frac {\frac {9 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}}{2}+3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}-\frac {510}{\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {573}{2}}{-\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {1870}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}+2 \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}+\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}\right )^{2}} \\ 1 \end {array}\right ]\)

\(\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\)	\(1\)	\(1\)	No	\(\left [\begin {array}{c} -\frac {135 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {935}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}+\frac {11 i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}+2 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {8}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {\frac {9 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}}{2}-\frac {3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{2}+\frac {255}{\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {573}{2}-\frac {9 i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}}{\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {935}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}+\frac {11 i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}+2 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\)

\(\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\)	\(1\)	\(1\)	No	\(\left [\begin {array}{c} -\frac {135 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {935}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}-\frac {11 i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}+2 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {8}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {\frac {9 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}}{2}-\frac {3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{2}+\frac {255}{\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {573}{2}+\frac {9 i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}}{\frac {11 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {935}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}+\frac {412}{3}-\frac {11 i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}+2 \left (\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{6}-\frac {85}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}{3}-\frac {170}{3 \left (4726+306 \sqrt {291}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\)

9.3 problem 3

9.3.1 Solution using Matrix exponential method

9.3.2 Solution using explicit Eigenvalue and Eigenvector method

9.3.3 Maple step by step solution