Internal problem ID [6716]
Internal file name [OUTPUT/5964_Sunday_June_05_2022_04_05_01_PM_1900378/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: 6.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "system of linear ODEs"
Solve \begin {align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )+4 y+2 \,{\mathrm e}^{-t} \sin \left (t \right ) \cos \left (t \right )\\ y^{\prime }&=5 x \left (t \right )+9 z \left (t \right )+8 \,{\mathrm e}^{-t} \cos \left (t \right )^{2}-4 \,{\mathrm e}^{-t}\\ z^{\prime }\left (t \right )&=y+6 z \left (t \right )-{\mathrm e}^{-t} \end {align*}
In this method, we will assume we have found the matrix exponential \(e^{A t}\) allready. There are different methods to determine this but will not be shown here. This is a system of linear ODE’s given as Warning. Unable to find the matrix exponential.
This is a system of linear ODE’s given as \begin {align*} \vec {x}'(t) &= A\, \vec {x}(t) + \vec {G}(t) \end {align*}
Or \begin {align*} \left [\begin {array}{c} x^{\prime }\left (t \right ) \\ y^{\prime } \\ z^{\prime }\left (t \right ) \end {array}\right ] &= \left [\begin {array}{ccc} -3 & 4 & 0 \\ 5 & 0 & 9 \\ 0 & 1 & 6 \end {array}\right ]\, \left [\begin {array}{c} x \left (t \right ) \\ y \\ z \left (t \right ) \end {array}\right ] + \left [\begin {array}{c} 2 \,{\mathrm e}^{-t} \sin \left (t \right ) \cos \left (t \right ) \\ 8 \,{\mathrm e}^{-t} \cos \left (t \right )^{2}-4 \,{\mathrm e}^{-t} \\ -{\mathrm e}^{-t} \end {array}\right ] \end {align*}
Since the system is nonhomogeneous, then the solution is given by \begin {align*} \vec {x}(t) &= \vec {x}_h(t) + \vec {x}_p(t) \end {align*}
Where \(\vec {x}_h(t)\) is the homogeneous solution to \(\vec {x}'(t) = A\, \vec {x}(t)\) and \(\vec {x}_p(t)\) is a particular solution to \(\vec {x}'(t) = A\, \vec {x}(t) + \vec {G}(t)\). The particular solution will be found using variation of parameters method applied to the fundamental matrix.
The first step is find the homogeneous solution. We start by finding the eigenvalues of \(A\). This is done by solving the following equation for the eigenvalues \(\lambda \) \begin {align*} \operatorname {det} \left ( A- \lambda I \right ) &= 0 \end {align*}
Expanding gives \begin {align*} \operatorname {det} \left (\left [\begin {array}{ccc} -3 & 4 & 0 \\ 5 & 0 & 9 \\ 0 & 1 & 6 \end {array}\right ]-\lambda \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) &= 0 \end {align*}
Therefore \begin {align*} \operatorname {det} \left (\left [\begin {array}{ccc} -3-\lambda & 4 & 0 \\ 5 & -\lambda & 9 \\ 0 & 1 & 6-\lambda \end {array}\right ]\right ) &= 0 \end {align*}
Which gives the characteristic equation \begin {align*} \lambda ^{3}-3 \lambda ^{2}-47 \lambda +93&=0 \end {align*}
The roots of the above are the eigenvalues. \begin {align*} \lambda _1 &= \frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1\\ \lambda _2 &= -\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\\ \lambda _3 &= -\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}
This table summarises the above result
| eigenvalue | algebraic multiplicity | type of eigenvalue |
| \(\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1\) | \(1\) | complex eigenvalue |
| \(-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\) | \(1\) | complex eigenvalue |
| \(-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\) | \(1\) | complex eigenvalue |
Now the eigenvector for each eigenvalue are found.
Considering the eigenvalue \(\lambda _{1} = \frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1\)
We need to solve \(A \vec {v} = \lambda \vec {v}\) or \((A-\lambda I) \vec {v} = \vec {0}\) which becomes \begin {align*} \left (\left [\begin {array}{ccc} -3 & 4 & 0 \\ 5 & 0 & 9 \\ 0 & 1 & 6 \end {array}\right ] - \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1\right ) \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccc} \frac {-\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}-12 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}-150}{3 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}} & 4 & 0 \\ 5 & -\frac {\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}}{3}+\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}} & 9 \\ 0 & 1 & \frac {-\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+15 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}-150}{3 \left (-594+6 i \sqrt {83949}\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 \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -4-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}&4&0&0\\ 5&-1-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}&9&0\\ 0&1&5-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}-\frac {5 R_{1}}{-4-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {-\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}-12 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}-150}{3 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}&4&0&0\\ 0&-\frac {2 \left (\left (i \sqrt {83949}+276\right ) \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+15 i \sqrt {83949}+26 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+2265\right )}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}} \left (\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+12 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+150\right )}&9&0\\ 0&1&\frac {-\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+15 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}-150}{3 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}+\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}} \left (\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+12 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+150\right ) R_{2}}{2 \left (i \sqrt {83949}+276\right ) \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+30 i \sqrt {83949}+52 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+4530} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {-\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}-12 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}-150}{3 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}&4&0&0\\ 0&-\frac {2 \left (\left (i \sqrt {83949}+276\right ) \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+15 i \sqrt {83949}+26 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+2265\right )}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}} \left (\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+12 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+150\right )}&9&0\\ 0&0&0&0 \end {array} \right ] \end {align*}
Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} \frac {-\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}-12 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}-150}{3 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}} & 4 & 0 \\ 0 & -\frac {2 \left (\left (i \sqrt {83949}+276\right ) \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+15 i \sqrt {83949}+26 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+2265\right )}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}} \left (\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+12 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+150\right )} & 9 \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] 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 \(\left \{v_{1} = \frac {324 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}} t \left (i \sqrt {83949}+2 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+25 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}-99\right )}{\left (i \sqrt {83949}\, \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+15 i \sqrt {83949}+26 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+276 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+2265\right ) \left (\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+12 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+150\right )}, v_{2} = \frac {27 t \left (i \sqrt {83949}+2 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+25 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}-99\right )}{i \sqrt {83949}\, \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+15 i \sqrt {83949}+26 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+276 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+2265}\right \}\)
Hence the solution is \[ \left [\begin {array}{c} \frac {324 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}} t \left (\operatorname {I} \sqrt {83949}+2 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {2}{3}}+25 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}-99\right )}{\left (\operatorname {I} \sqrt {83949}\, \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}+15 \,\operatorname {I} \sqrt {83949}+26 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {2}{3}}+276 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}+2265\right ) \left (\left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {2}{3}}+12 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}+150\right )} \\ \frac {27 t \left (\operatorname {I} \sqrt {83949}+2 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {2}{3}}+25 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}-99\right )}{\operatorname {I} \sqrt {83949}\, \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}+15 \,\operatorname {I} \sqrt {83949}+26 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {2}{3}}+276 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}+2265} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {324 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}} t \left (i \sqrt {83949}+2 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+25 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}-99\right )}{\left (i \sqrt {83949}\, \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+15 i \sqrt {83949}+26 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+276 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+2265\right ) \left (\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+12 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+150\right )} \\ \frac {27 t \left (i \sqrt {83949}+2 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+25 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}-99\right )}{i \sqrt {83949}\, \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+15 i \sqrt {83949}+26 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+276 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+2265} \\ 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 {324 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}} t \left (\operatorname {I} \sqrt {83949}+2 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {2}{3}}+25 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}-99\right )}{\left (\operatorname {I} \sqrt {83949}\, \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}+15 \,\operatorname {I} \sqrt {83949}+26 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {2}{3}}+276 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}+2265\right ) \left (\left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {2}{3}}+12 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}+150\right )} \\ \frac {27 t \left (\operatorname {I} \sqrt {83949}+2 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {2}{3}}+25 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}-99\right )}{\operatorname {I} \sqrt {83949}\, \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}+15 \,\operatorname {I} \sqrt {83949}+26 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {2}{3}}+276 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}+2265} \\ t \end {array}\right ] = t \left [\begin {array}{c} \frac {324 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}} \left (i \sqrt {83949}+2 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+25 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}-99\right )}{\left (i \sqrt {83949}\, \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+15 i \sqrt {83949}+26 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+276 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+2265\right ) \left (\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+12 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+150\right )} \\ \frac {27 i \sqrt {83949}+54 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+675 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}-2673}{i \sqrt {83949}\, \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+15 i \sqrt {83949}+26 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+276 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+2265} \\ 1 \end {array}\right ] \] Let \(t = 1\) the eigenvector becomes \[ \left [\begin {array}{c} \frac {324 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}} t \left (\operatorname {I} \sqrt {83949}+2 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {2}{3}}+25 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}-99\right )}{\left (\operatorname {I} \sqrt {83949}\, \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}+15 \,\operatorname {I} \sqrt {83949}+26 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {2}{3}}+276 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}+2265\right ) \left (\left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {2}{3}}+12 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}+150\right )} \\ \frac {27 t \left (\operatorname {I} \sqrt {83949}+2 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {2}{3}}+25 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}-99\right )}{\operatorname {I} \sqrt {83949}\, \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}+15 \,\operatorname {I} \sqrt {83949}+26 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {2}{3}}+276 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}+2265} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {324 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}} \left (i \sqrt {83949}+2 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+25 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}-99\right )}{\left (i \sqrt {83949}\, \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+15 i \sqrt {83949}+26 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+276 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+2265\right ) \left (\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+12 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+150\right )} \\ \frac {27 i \sqrt {83949}+54 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+675 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}-2673}{i \sqrt {83949}\, \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+15 i \sqrt {83949}+26 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+276 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+2265} \\ 1 \end {array}\right ] \] Which is normalized to \[ \left [\begin {array}{c} \frac {324 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}} t \left (\operatorname {I} \sqrt {83949}+2 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {2}{3}}+25 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}-99\right )}{\left (\operatorname {I} \sqrt {83949}\, \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}+15 \,\operatorname {I} \sqrt {83949}+26 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {2}{3}}+276 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}+2265\right ) \left (\left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {2}{3}}+12 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}+150\right )} \\ \frac {27 t \left (\operatorname {I} \sqrt {83949}+2 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {2}{3}}+25 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}-99\right )}{\operatorname {I} \sqrt {83949}\, \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}+15 \,\operatorname {I} \sqrt {83949}+26 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {2}{3}}+276 \left (-594+6 \,\operatorname {I} \sqrt {83949}\right )^{\frac {1}{3}}+2265} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {324 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}} \left (i \sqrt {83949}+2 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+25 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}-99\right )}{\left (i \sqrt {83949}\, \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+15 i \sqrt {83949}+26 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+276 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+2265\right ) \left (\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+12 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+150\right )} \\ \frac {27 i \sqrt {83949}+54 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+675 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}-2673}{i \sqrt {83949}\, \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+15 i \sqrt {83949}+26 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}+276 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+2265} \\ 1 \end {array}\right ] \] Considering the eigenvalue \(\lambda _{2} = -\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\)
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 & 0 \\ 5 & 0 & 9 \\ 0 & 1 & 6 \end {array}\right ] - \left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\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} -4+\frac {5 \sqrt {2}\, \sqrt {3}\, \cos \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )}{3}-5 \sqrt {2}\, \sin \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right ) & 4 & 0 \\ 5 & -1+\frac {5 \sqrt {2}\, \sqrt {3}\, \cos \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )}{3}-5 \sqrt {2}\, \sin \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right ) & 9 \\ 0 & 1 & 5+\frac {5 \sqrt {2}\, \sqrt {3}\, \cos \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )}{3}-5 \sqrt {2}\, \sin \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right ) \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \end {align*}
Now forward elimination is applied to solve for the eigenvector \(\vec {v}\). The augmented matrix is \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -4+\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}+\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}&4&0&0\\ 5&-1+\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}+\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}&9&0\\ 0&1&5+\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}+\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}-\frac {5 R_{1}}{-4+\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}+\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -4+\frac {5 \sqrt {2}\, \sqrt {3}\, \cos \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )}{3}-5 \sqrt {2}\, \sin \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )&4&0&0\\ 0&\frac {2 \left (276 i \sqrt {3}-i \sqrt {83949}-3 \sqrt {27983}-276\right ) \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}-4530 i \sqrt {3}-30 i \sqrt {83949}+90 \sqrt {27983}+104 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}-4530}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}} \left (150-24 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+i \left (\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}-150\right ) \sqrt {3}+\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}\right )}&9&0\\ 0&1&5+\frac {5 \sqrt {2}\, \sqrt {3}\, \cos \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )}{3}-5 \sqrt {2}\, \sin \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}} \left (150-24 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+i \left (\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}-150\right ) \sqrt {3}+\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}\right ) R_{2}}{2 \left (\left (276 i \sqrt {3}-i \sqrt {83949}-3 \sqrt {27983}-276\right ) \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}-2265 i \sqrt {3}-15 i \sqrt {83949}+45 \sqrt {27983}+52 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}-2265\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -4+\frac {5 \sqrt {2}\, \sqrt {3}\, \cos \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )}{3}-5 \sqrt {2}\, \sin \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )&4&0&0\\ 0&\frac {2 \left (276 i \sqrt {3}-i \sqrt {83949}-3 \sqrt {27983}-276\right ) \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}-4530 i \sqrt {3}-30 i \sqrt {83949}+90 \sqrt {27983}+104 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}-4530}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}} \left (150-24 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+i \left (\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}-150\right ) \sqrt {3}+\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}\right )}&9&0\\ 0&0&\frac {8652+5 \left (2265 i+\left (\left (\sqrt {27983}+92\right ) \sqrt {3}+i \sqrt {27983}-276 i\right ) \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}-\frac {52 \sqrt {3}\, \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}}{3}+5 \left (-3 \sqrt {27983}+151\right ) \sqrt {3}+15 i \sqrt {27983}\right ) \sqrt {2}\, \cos \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )+5 \left (-2265-\left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \left (276+i \left (\sqrt {27983}-276\right ) \sqrt {3}+3 \sqrt {27983}\right )+52 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}-15 i \left (\sqrt {27983}+151\right ) \sqrt {3}+45 \sqrt {27983}\right ) \sqrt {2}\, \sin \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )+5 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \left (411+i \left (-411+\sqrt {27983}\right ) \sqrt {3}+3 \sqrt {27983}\right )-368 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+6 i \left (1442+17 \sqrt {27983}\right ) \sqrt {3}-306 \sqrt {27983}}{2265+\left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \left (276+i \left (\sqrt {27983}-276\right ) \sqrt {3}+3 \sqrt {27983}\right )-52 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+15 i \left (\sqrt {27983}+151\right ) \sqrt {3}-45 \sqrt {27983}}&0 \end {array} \right ] \end {align*}
Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} -4+\frac {5 \sqrt {2}\, \sqrt {3}\, \cos \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )}{3}-5 \sqrt {2}\, \sin \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right ) & 4 & 0 \\ 0 & \frac {2 \left (276 i \sqrt {3}-i \sqrt {83949}-3 \sqrt {27983}-276\right ) \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}-4530 i \sqrt {3}-30 i \sqrt {83949}+90 \sqrt {27983}+104 \left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}-4530}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}} \left (150-24 \left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+i \left (\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}-150\right ) \sqrt {3}+\left (-594+6 i \sqrt {83949}\right )^{\frac {2}{3}}\right )} & 9 \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] 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 \(\left \{v_{1} = \frac {162 t \left (i \sqrt {3}\, \sqrt {27983}-25 i \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}-99 i \sqrt {3}-4 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}-3 \sqrt {27983}+25 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}-99\right )}{\left (5 \sqrt {3}\, \sqrt {2}\, \sin \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}\right )+6\right ) \left (i \sqrt {27983}\, \sqrt {3}\, \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+15 i \sqrt {3}\, \sqrt {27983}-276 i \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}+3 \sqrt {27983}\, \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+2265 i \sqrt {3}-52 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}-45 \sqrt {27983}+276 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+2265\right )}, v_{2} = \frac {27 t \left (i \sqrt {3}\, \sqrt {27983}-25 i \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}-99 i \sqrt {3}-4 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}-3 \sqrt {27983}+25 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}-99\right )}{i \sqrt {27983}\, \sqrt {3}\, \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+15 i \sqrt {3}\, \sqrt {27983}-276 i \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}+3 \sqrt {27983}\, \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+2265 i \sqrt {3}-52 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}-45 \sqrt {27983}+276 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+2265}\right \}\)
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 {162 t \left (\operatorname {I} \sqrt {3}\, \sqrt {27983}-25 \,\operatorname {I} \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}-99 \,\operatorname {I} \sqrt {3}-4 \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}-3 \sqrt {27983}+25 \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}-99\right )}{\left (5 \sqrt {3}\, \sqrt {2}\, \sin \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}\right )+6\right ) \left (\operatorname {I} \sqrt {27983}\, \sqrt {3}\, \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+15 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}-276 \,\operatorname {I} \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}+3 \sqrt {27983}\, \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+2265 \,\operatorname {I} \sqrt {3}-52 \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}-45 \sqrt {27983}+276 \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+2265\right )} \\ \frac {27 t \left (\operatorname {I} \sqrt {3}\, \sqrt {27983}-25 \,\operatorname {I} \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}-99 \,\operatorname {I} \sqrt {3}-4 \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}-3 \sqrt {27983}+25 \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}-99\right )}{\operatorname {I} \sqrt {27983}\, \sqrt {3}\, \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+15 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}-276 \,\operatorname {I} \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}+3 \sqrt {27983}\, \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+2265 \,\operatorname {I} \sqrt {3}-52 \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}-45 \sqrt {27983}+276 \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+2265} \\ t \end {array}\right ] = t \left [\begin {array}{c} \frac {162 i \sqrt {3}\, \sqrt {27983}-4050 i \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}-16038 i \sqrt {3}-648 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}-486 \sqrt {27983}+4050 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}-16038}{\left (5 \sqrt {3}\, \sqrt {2}\, \sin \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}\right )+6\right ) \left (i \sqrt {27983}\, \sqrt {3}\, \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+15 i \sqrt {3}\, \sqrt {27983}-276 i \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}+3 \sqrt {27983}\, \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+2265 i \sqrt {3}-52 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}-45 \sqrt {27983}+276 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+2265\right )} \\ \frac {27 i \sqrt {3}\, \sqrt {27983}-675 i \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}-2673 i \sqrt {3}-108 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}-81 \sqrt {27983}+675 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}-2673}{i \sqrt {27983}\, \sqrt {3}\, \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+15 i \sqrt {3}\, \sqrt {27983}-276 i \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}+3 \sqrt {27983}\, \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+2265 i \sqrt {3}-52 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}-45 \sqrt {27983}+276 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+2265} \\ 1 \end {array}\right ] \] Let \(t = 1\) the eigenvector becomes \[ \text {Expression too large to display} \] Which is normalized to \[ \text {Expression too large to display} \] Considering the eigenvalue \(\lambda _{3} = -\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\)
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 & 0 \\ 5 & 0 & 9 \\ 0 & 1 & 6 \end {array}\right ] - \left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\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} -4+\frac {5 \sqrt {2}\, \sqrt {3}\, \cos \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )}{3}+5 \sqrt {2}\, \sin \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right ) & 4 & 0 \\ 5 & -1+\frac {5 \sqrt {2}\, \sqrt {3}\, \cos \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )}{3}+5 \sqrt {2}\, \sin \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right ) & 9 \\ 0 & 1 & 5+\frac {5 \sqrt {2}\, \sqrt {3}\, \cos \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )}{3}+5 \sqrt {2}\, \sin \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right ) \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \end {align*}
Now forward elimination is applied to solve for the eigenvector \(\vec {v}\). The augmented matrix is \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -4+\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}+\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}&4&0&0\\ 5&-1+\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}+\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}&9&0\\ 0&1&5+\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}+\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}&0 \end {array} \right ] \] \begin {align*} R_{2} = R_{2}-\frac {5 R_{1}}{-4+\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}+\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -4+\frac {5 \sqrt {2}\, \sqrt {3}\, \cos \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )}{3}+5 \sqrt {2}\, \sin \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )&4&0&0\\ 0&\frac {4530+2 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \left (276+i \left (\sqrt {27983}+276\right ) \sqrt {3}-3 \sqrt {27983}\right )-104 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+30 i \left (-151+\sqrt {27983}\right ) \sqrt {3}+90 \sqrt {27983}}{\left (i \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}} \sqrt {3}-\left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}-150 i \sqrt {3}+24 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}-150\right ) \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}}&9&0\\ 0&1&5+\frac {5 \sqrt {2}\, \sqrt {3}\, \cos \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )}{3}+5 \sqrt {2}\, \sin \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}-\frac {\left (i \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}} \sqrt {3}-\left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}-150 i \sqrt {3}+24 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}-150\right ) \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} R_{2}}{2 \left (2265+\left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \left (276+i \left (\sqrt {27983}+276\right ) \sqrt {3}-3 \sqrt {27983}\right )-52 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+15 i \left (-151+\sqrt {27983}\right ) \sqrt {3}+45 \sqrt {27983}\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -4+\frac {5 \sqrt {2}\, \sqrt {3}\, \cos \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )}{3}+5 \sqrt {2}\, \sin \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )&4&0&0\\ 0&\frac {4530+2 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \left (276+i \left (\sqrt {27983}+276\right ) \sqrt {3}-3 \sqrt {27983}\right )-104 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+30 i \left (-151+\sqrt {27983}\right ) \sqrt {3}+90 \sqrt {27983}}{\left (i \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}} \sqrt {3}-\left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}-150 i \sqrt {3}+24 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}-150\right ) \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}}&9&0\\ 0&0&\frac {8652+5 \left (-2265 i+\left (\left (-\sqrt {27983}+92\right ) \sqrt {3}+i \sqrt {27983}+276 i\right ) \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}-\frac {52 \sqrt {3}\, \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}}{3}+5 \left (3 \sqrt {27983}+151\right ) \sqrt {3}+15 i \sqrt {27983}\right ) \sqrt {2}\, \cos \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )+5 \sqrt {2}\, \sin \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right ) \left (2265+\left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \left (276+i \left (\sqrt {27983}+276\right ) \sqrt {3}-3 \sqrt {27983}\right )-52 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+15 i \left (-151+\sqrt {27983}\right ) \sqrt {3}+45 \sqrt {27983}\right )+5 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \left (411+i \left (\sqrt {27983}+411\right ) \sqrt {3}-3 \sqrt {27983}\right )-368 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+6 i \left (-1442+17 \sqrt {27983}\right ) \sqrt {3}+306 \sqrt {27983}}{2265+\left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \left (276+i \left (\sqrt {27983}+276\right ) \sqrt {3}-3 \sqrt {27983}\right )-52 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+15 i \left (-151+\sqrt {27983}\right ) \sqrt {3}+45 \sqrt {27983}}&0 \end {array} \right ] \end {align*}
Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} -4+\frac {5 \sqrt {2}\, \sqrt {3}\, \cos \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right )}{3}+5 \sqrt {2}\, \sin \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}+\frac {\pi }{6}\right ) & 4 & 0 \\ 0 & \frac {4530+2 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \left (276+i \left (\sqrt {27983}+276\right ) \sqrt {3}-3 \sqrt {27983}\right )-104 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+30 i \left (-151+\sqrt {27983}\right ) \sqrt {3}+90 \sqrt {27983}}{\left (i \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}} \sqrt {3}-\left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}-150 i \sqrt {3}+24 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}-150\right ) \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}} & 9 \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] 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 \(\left \{v_{1} = -\frac {324 \left (i \sqrt {3}\, \sqrt {27983}+25 i \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}+99 i \sqrt {3}-4 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+3 \sqrt {27983}+25 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}-99\right ) t}{\left (5 \sqrt {3}\, \sqrt {2}\, \sin \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}\right )+15 \sqrt {2}\, \cos \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}\right )-12\right ) \left (-3 \sqrt {27983}\, \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+i \sqrt {27983}\, \sqrt {3}\, \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+15 i \sqrt {3}\, \sqrt {27983}+276 i \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}-2265 i \sqrt {3}-52 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+45 \sqrt {27983}+276 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+2265\right )}, v_{2} = \frac {27 t \left (i \sqrt {3}\, \sqrt {27983}+25 i \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}+99 i \sqrt {3}-4 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+3 \sqrt {27983}+25 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}-99\right )}{-3 \sqrt {27983}\, \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+i \sqrt {27983}\, \sqrt {3}\, \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+15 i \sqrt {3}\, \sqrt {27983}+276 i \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}-2265 i \sqrt {3}-52 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+45 \sqrt {27983}+276 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+2265}\right \}\)
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 {324 \left (\operatorname {I} \sqrt {3}\, \sqrt {27983}+25 \,\operatorname {I} \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}+99 \,\operatorname {I} \sqrt {3}-4 \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+3 \sqrt {27983}+25 \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}-99\right ) t}{\left (5 \sqrt {3}\, \sqrt {2}\, \sin \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}\right )+15 \sqrt {2}\, \cos \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}\right )-12\right ) \left (-3 \sqrt {27983}\, \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+\operatorname {I} \sqrt {27983}\, \sqrt {3}\, \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+15 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}+276 \,\operatorname {I} \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}-2265 \,\operatorname {I} \sqrt {3}-52 \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+45 \sqrt {27983}+276 \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+2265\right )} \\ \frac {27 t \left (\operatorname {I} \sqrt {3}\, \sqrt {27983}+25 \,\operatorname {I} \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}+99 \,\operatorname {I} \sqrt {3}-4 \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+3 \sqrt {27983}+25 \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}-99\right )}{-3 \sqrt {27983}\, \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+\operatorname {I} \sqrt {27983}\, \sqrt {3}\, \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+15 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}+276 \,\operatorname {I} \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}-2265 \,\operatorname {I} \sqrt {3}-52 \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+45 \sqrt {27983}+276 \left (-594+6 \,\operatorname {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+2265} \\ t \end {array}\right ] = t \left [\begin {array}{c} -\frac {324 \left (i \sqrt {3}\, \sqrt {27983}+25 i \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}+99 i \sqrt {3}-4 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+3 \sqrt {27983}+25 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}-99\right )}{\left (5 \sqrt {3}\, \sqrt {2}\, \sin \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}\right )+15 \sqrt {2}\, \cos \left (\frac {\arctan \left (\frac {33 \sqrt {83949}}{12358}\right )}{6}\right )-12\right ) \left (-3 \sqrt {27983}\, \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+i \sqrt {27983}\, \sqrt {3}\, \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+15 i \sqrt {3}\, \sqrt {27983}+276 i \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}-2265 i \sqrt {3}-52 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+45 \sqrt {27983}+276 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+2265\right )} \\ \frac {27 i \sqrt {3}\, \sqrt {27983}+675 i \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}+2673 i \sqrt {3}-108 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+81 \sqrt {27983}+675 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}-2673}{-3 \sqrt {27983}\, \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+i \sqrt {27983}\, \sqrt {3}\, \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+15 i \sqrt {3}\, \sqrt {27983}+276 i \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}} \sqrt {3}-2265 i \sqrt {3}-52 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+45 \sqrt {27983}+276 \left (-594+6 i \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+2265} \\ 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 |
| \(\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1\) | \(1\) | \(1\) | No | \(\left [\begin {array}{c} \frac {324}{\left (\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+\frac {150}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}-26+\left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1\right )^{2}\right ) \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+4\right )} \\ \frac {81}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+\frac {150}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}-26+\left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1\right )^{2}} \\ 1 \end {array}\right ]\) |
| \(-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\) | \(1\) | \(1\) | No | \(\left [\begin {array}{c} \frac {324}{\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}-26+\frac {3 i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+4+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {81}{-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}-26+\frac {3 i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\) |
| \(-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\) | \(1\) | \(1\) | No | \(\left [\begin {array}{c} \frac {324}{\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}-26-\frac {3 i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+4-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {81}{-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}-26-\frac {3 i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\) |
Now that we found the eigenvalues and associated eigenvectors, we will go over each eigenvalue and generate the solution basis. The only problem we need to take care of is if the eigenvalue is defective. Therefore the homogeneous solution is \begin {align*} \vec {x}_h(t) &= c_{1} \vec {x}_{1}(t) + c_{2} \vec {x}_{2}(t) + c_{3} \vec {x}_{3}(t) \end {align*}
Which is written as \begin {align*} \left [\begin {array}{c} x \left (t \right ) \\ y \\ z \left (t \right ) \end {array}\right ] &= c_{1} \left [\begin {array}{c} \frac {324 \,{\mathrm e}^{\left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1\right ) t}}{\left (\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+\frac {150}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}-26+\left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1\right )^{2}\right ) \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+4\right )} \\ \frac {81 \,{\mathrm e}^{\left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1\right ) t}}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+\frac {150}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}-26+\left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1\right )^{2}} \\ {\mathrm e}^{\left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1\right ) t} \end {array}\right ] + c_{2} \left [\begin {array}{c} \frac {324 \,{\mathrm e}^{\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}}{\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}-26+\frac {3 i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+4+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {81 \,{\mathrm e}^{\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}}{-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}-26+\frac {3 i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ {\mathrm e}^{\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} \end {array}\right ] + c_{3} \left [\begin {array}{c} \frac {324 \,{\mathrm e}^{\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}}{\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}-26-\frac {3 i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+4-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {81 \,{\mathrm e}^{\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}}{-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}-26-\frac {3 i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ {\mathrm e}^{\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} \end {array}\right ] \end {align*}
Now that we found homogeneous solution above, we need to find a particular solution \(\vec {x}_p(t)\). We will use Variation of parameters. The fundamental matrix is \[ \Phi =\begin {bmatrix} \vec {x}_{1} & \vec {x}_{2} & \cdots \end {bmatrix} \] Where \(\vec {x}_i\) are the solution basis found above. Therefore the fundamental matrix is \begin {align*} \Phi (t)&= \text {Expression too large to display} \end {align*}
The particular solution is then given by \begin {align*} \vec {x}_p(t) &= \Phi \int { \Phi ^{-1} \vec {G}(t) \, dt}\\ \end {align*}
But \begin {align*} \Phi ^{-1} &= \text {Expression too large to display} \end {align*}
Hence \begin {align*} \vec {x}_p(t) &= \text {Expression too large to display} \int { \text {Expression too large to display} \left [\begin {array}{c} 2 \,{\mathrm e}^{-t} \sin \left (t \right ) \cos \left (t \right ) \\ 8 \,{\mathrm e}^{-t} \cos \left (t \right )^{2}-4 \,{\mathrm e}^{-t} \\ -{\mathrm e}^{-t} \end {array}\right ] \, dt}\\ &= \text {Expression too large to display} \int { \text {Expression too large to display} \, dt}\\ &= \text {Expression too large to display} \text {Expression too large to display} \\ &= \text {Expression too large to display} \end {align*}
Now that we found particular solution, the final solution is \begin {align*} \vec {x}(t) &= \vec {x}_h(t) + \vec {x}_p(t)\\ \left [\begin {array}{c} x \left (t \right ) \\ y \\ z \left (t \right ) \end {array}\right ] &= \left [\begin {array}{c} \frac {324 c_{1} {\mathrm e}^{\left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1\right ) t}}{\left (\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+\frac {150}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}-26+\left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1\right )^{2}\right ) \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+4\right )} \\ \frac {81 c_{1} {\mathrm e}^{\left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1\right ) t}}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}+\frac {150}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}-26+\left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1\right )^{2}} \\ c_{1} {\mathrm e}^{\left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1\right ) t} \end {array}\right ] + \left [\begin {array}{c} \frac {324 c_{2} {\mathrm e}^{\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}}{\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}-26+\frac {3 i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+4+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {81 c_{2} {\mathrm e}^{\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}}{-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}-26+\frac {3 i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ c_{2} {\mathrm e}^{\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} \end {array}\right ] + \left [\begin {array}{c} \frac {324 c_{3} {\mathrm e}^{\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}}{\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}-26-\frac {3 i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+4-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {81 c_{3} {\mathrm e}^{\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}}{-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}-26-\frac {3 i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ c_{3} {\mathrm e}^{\left (-\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 i \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} \end {array}\right ] + \text {Expression too large to display} \end {align*}
Which becomes \begin {align*} \text {Expression too large to display} \end {align*}
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [x^{\prime }\left (t \right )=-3 x \left (t \right )+4 y+\frac {2 \sin \left (t \right ) \cos \left (t \right )}{{\mathrm e}^{t}}, y^{\prime }=5 x \left (t \right )+9 z \left (t \right )+\frac {8 \cos \left (t \right )^{2}}{{\mathrm e}^{t}}-\frac {4}{{\mathrm e}^{t}}, z^{\prime }\left (t \right )=y+6 z \left (t \right )-\frac {1}{{\mathrm e}^{t}}\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 & 0 \\ 5 & 0 & 9 \\ 0 & 1 & 6 \end {array}\right ]\cdot {\moverset {\rightarrow }{x}}\left (t \right )+\left [\begin {array}{c} -\frac {3 x \left (t \right ) {\mathrm e}^{t}-4 y \,{\mathrm e}^{t}-2 \sin \left (t \right ) \cos \left (t \right )}{{\mathrm e}^{t}}+3 x \left (t \right )-4 y \\ \frac {9 z \left (t \right ) {\mathrm e}^{t}+5 x \left (t \right ) {\mathrm e}^{t}+8 \cos \left (t \right )^{2}-4}{{\mathrm e}^{t}}-5 x \left (t \right )-9 z \left (t \right ) \\ \frac {6 z \left (t \right ) {\mathrm e}^{t}+y \,{\mathrm e}^{t}-1}{{\mathrm e}^{t}}-y-6 z \left (t \right ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}^{\prime }\left (t \right )=\left [\begin {array}{ccc} -3 & 4 & 0 \\ 5 & 0 & 9 \\ 0 & 1 & 6 \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 & 0 \\ 5 & 0 & 9 \\ 0 & 1 & 6 \end {array}\right ] \\ \bullet & {} & \textrm {Rewrite the system as}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}^{\prime }\left (t \right )=A \cdot {\moverset {\rightarrow }{x}}\left (t \right ) \\ \bullet & {} & \textrm {To solve the system, find the eigenvalues and eigenvectors of}\hspace {3pt} A \\ \bullet & {} & \textrm {Eigenpairs of}\hspace {3pt} A \\ {} & {} & \left [\left [\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1, \left [\begin {array}{c} \frac {324}{\left (\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}+\frac {150}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26+\left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1\right )^{2}\right ) \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+4\right )} \\ \frac {81}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}+\frac {150}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26+\left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1\right )^{2}} \\ 1 \end {array}\right ]\right ], \left [-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {324}{\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26-\frac {3 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+4-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {81}{-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26-\frac {3 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\right ], \left [-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {324}{\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26+\frac {3 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+4+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {81}{-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26+\frac {3 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1, \left [\begin {array}{c} \frac {324}{\left (\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}+\frac {150}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26+\left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1\right )^{2}\right ) \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+4\right )} \\ \frac {81}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}+\frac {150}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26+\left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1\right )^{2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}_{1}={\mathrm e}^{\left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1\right ) t}\cdot \left [\begin {array}{c} \frac {324}{\left (\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}+\frac {150}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26+\left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1\right )^{2}\right ) \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+4\right )} \\ \frac {81}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}+\frac {150}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26+\left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1\right )^{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {324}{\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26-\frac {3 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+4-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {81}{-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26-\frac {3 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}_{2}={\mathrm e}^{\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}\cdot \left [\begin {array}{c} \frac {324}{\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26-\frac {3 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+4-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {81}{-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26-\frac {3 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {324}{\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26+\frac {3 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+4+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {81}{-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26+\frac {3 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}_{3}={\mathrm e}^{\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}\cdot \left [\begin {array}{c} \frac {324}{\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26+\frac {3 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+4+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {81}{-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26+\frac {3 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {General solution to the system of ODEs}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}=c_{1} {\moverset {\rightarrow }{x}}_{1}+c_{2} {\moverset {\rightarrow }{x}}_{2}+c_{3} {\moverset {\rightarrow }{x}}_{3} \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{x}}=c_{1} {\mathrm e}^{\left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1\right ) t}\cdot \left [\begin {array}{c} \frac {324}{\left (\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}+\frac {150}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26+\left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1\right )^{2}\right ) \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+4\right )} \\ \frac {81}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}+\frac {150}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26+\left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}+\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1\right )^{2}} \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}\cdot \left [\begin {array}{c} \frac {324}{\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26-\frac {3 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+4-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {81}{-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26-\frac {3 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ]+c_{3} {\mathrm e}^{\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}\cdot \left [\begin {array}{c} \frac {324}{\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26+\frac {3 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}\right ) \left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+4+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )} \\ \frac {81}{-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{2}-\frac {75}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}-26+\frac {3 \,\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}+\left (-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{6}-\frac {25}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}+1+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}{3}-\frac {50}{\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Substitute in vector of dependent variables}\hspace {3pt} \\ {} & {} & \left [\begin {array}{c} x \left (t \right ) \\ y \\ z \left (t \right ) \end {array}\right ]=\left [\begin {array}{c} -\frac {977724 \left (\frac {\sqrt {3}\, \sqrt {27983}}{99}+\mathrm {I}\right ) \left (-\frac {3 c_{2} \left (\left (\left (\mathrm {I}+\frac {\sqrt {3}}{3}\right ) \sqrt {27983}+\frac {823 \,\mathrm {I}}{6}-\frac {823 \sqrt {3}}{6}\right ) \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+\left (\left (\frac {\mathrm {I}}{12}-\frac {\sqrt {3}}{36}\right ) \sqrt {27983}-\frac {133 \,\mathrm {I}}{12}-\frac {133 \sqrt {3}}{12}\right ) \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+\frac {29 \sqrt {3}\, \sqrt {27983}}{3}+957 \,\mathrm {I}\right ) {\mathrm e}^{-\frac {5 \left (\sqrt {2}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )-3 \sqrt {2}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )-\frac {3}{5}\right ) t}{3}}}{823}+\frac {3 \left (\left (\left (\mathrm {I}-\frac {\sqrt {3}}{3}\right ) \sqrt {27983}-\frac {823 \,\mathrm {I}}{6}-\frac {823 \sqrt {3}}{6}\right ) \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+\left (\left (\frac {\mathrm {I}}{12}+\frac {\sqrt {3}}{36}\right ) \sqrt {27983}+\frac {133 \,\mathrm {I}}{12}-\frac {133 \sqrt {3}}{12}\right ) \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}-\frac {29 \sqrt {3}\, \sqrt {27983}}{3}-957 \,\mathrm {I}\right ) c_{3} {\mathrm e}^{-\frac {5 \left (\sqrt {2}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )+3 \sqrt {2}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )-\frac {3}{5}\right ) t}{3}}}{823}+{\mathrm e}^{\frac {\left (\left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+3 \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+150\right ) t}{3 \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}}} \left (\left (\mathrm {I}+\frac {2 \sqrt {3}\, \sqrt {27983}}{823}\right ) \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+\left (-\frac {133 \,\mathrm {I}}{1646}-\frac {\sqrt {3}\, \sqrt {27983}}{4938}\right ) \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}-\frac {29 \sqrt {3}\, \sqrt {27983}}{823}-\frac {2871 \,\mathrm {I}}{823}\right ) c_{1} \right )}{17820 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}+6673320} \\ \frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {2}{3}} \left (c_{2} {\mathrm e}^{-\frac {5 \left (\sqrt {2}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )-3 \sqrt {2}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )-\frac {3}{5}\right ) t}{3}} \left (150 \,\mathrm {I}+\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {2}{3}} \left (\mathrm {I}-\sqrt {3}\right )+150 \sqrt {3}+30 \,\mathrm {I} \left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}\right )+c_{3} {\mathrm e}^{-\frac {5 \left (\sqrt {2}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )+3 \sqrt {2}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )-\frac {3}{5}\right ) t}{3}} \left (150 \,\mathrm {I}+\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {2}{3}} \left (\sqrt {3}+\mathrm {I}\right )-150 \sqrt {3}+30 \,\mathrm {I} \left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}\right )+30 \,\mathrm {I} \left (\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {2}{3}}}{15}-10\right ) c_{1} {\mathrm e}^{\frac {\left (\left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+3 \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+150\right ) t}{3 \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}}}\right )}{36 \left (99 \,\mathrm {I}+\sqrt {83949}\right )} \\ {\mathrm e}^{\frac {\left (\left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+3 \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+150\right ) t}{3 \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}}} c_{1} +{\mathrm e}^{-\frac {5 \left (\sqrt {2}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )-3 \sqrt {2}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )-\frac {3}{5}\right ) t}{3}} c_{2} +c_{3} {\mathrm e}^{-\frac {5 \left (\sqrt {2}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )+3 \sqrt {2}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )-\frac {3}{5}\right ) t}{3}} \end {array}\right ] \\ \bullet & {} & \textrm {Solution to the system of ODEs}\hspace {3pt} \\ {} & {} & \left \{x \left (t \right )=-\frac {977724 \left (\frac {\sqrt {3}\, \sqrt {27983}}{99}+\mathrm {I}\right ) \left (-\frac {3 c_{2} \left (\left (\left (\mathrm {I}+\frac {\sqrt {3}}{3}\right ) \sqrt {27983}+\frac {823 \,\mathrm {I}}{6}-\frac {823 \sqrt {3}}{6}\right ) \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+\left (\left (\frac {\mathrm {I}}{12}-\frac {\sqrt {3}}{36}\right ) \sqrt {27983}-\frac {133 \,\mathrm {I}}{12}-\frac {133 \sqrt {3}}{12}\right ) \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+\frac {29 \sqrt {3}\, \sqrt {27983}}{3}+957 \,\mathrm {I}\right ) {\mathrm e}^{-\frac {5 \left (\sqrt {2}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )-3 \sqrt {2}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )-\frac {3}{5}\right ) t}{3}}}{823}+\frac {3 \left (\left (\left (\mathrm {I}-\frac {\sqrt {3}}{3}\right ) \sqrt {27983}-\frac {823 \,\mathrm {I}}{6}-\frac {823 \sqrt {3}}{6}\right ) \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+\left (\left (\frac {\mathrm {I}}{12}+\frac {\sqrt {3}}{36}\right ) \sqrt {27983}+\frac {133 \,\mathrm {I}}{12}-\frac {133 \sqrt {3}}{12}\right ) \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}-\frac {29 \sqrt {3}\, \sqrt {27983}}{3}-957 \,\mathrm {I}\right ) c_{3} {\mathrm e}^{-\frac {5 \left (\sqrt {2}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )+3 \sqrt {2}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )-\frac {3}{5}\right ) t}{3}}}{823}+{\mathrm e}^{\frac {\left (\left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+3 \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+150\right ) t}{3 \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}}} \left (\left (\mathrm {I}+\frac {2 \sqrt {3}\, \sqrt {27983}}{823}\right ) \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+\left (-\frac {133 \,\mathrm {I}}{1646}-\frac {\sqrt {3}\, \sqrt {27983}}{4938}\right ) \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}-\frac {29 \sqrt {3}\, \sqrt {27983}}{823}-\frac {2871 \,\mathrm {I}}{823}\right ) c_{1} \right )}{17820 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}+6673320}, y=\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {2}{3}} \left (c_{2} {\mathrm e}^{-\frac {5 \left (\sqrt {2}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )-3 \sqrt {2}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )-\frac {3}{5}\right ) t}{3}} \left (150 \,\mathrm {I}+\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {2}{3}} \left (\mathrm {I}-\sqrt {3}\right )+150 \sqrt {3}+30 \,\mathrm {I} \left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}\right )+c_{3} {\mathrm e}^{-\frac {5 \left (\sqrt {2}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )+3 \sqrt {2}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )-\frac {3}{5}\right ) t}{3}} \left (150 \,\mathrm {I}+\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {2}{3}} \left (\sqrt {3}+\mathrm {I}\right )-150 \sqrt {3}+30 \,\mathrm {I} \left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}\right )+30 \,\mathrm {I} \left (\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {1}{3}}-\frac {\left (-594+6 \,\mathrm {I} \sqrt {83949}\right )^{\frac {2}{3}}}{15}-10\right ) c_{1} {\mathrm e}^{\frac {\left (\left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+3 \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+150\right ) t}{3 \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}}}\right )}{36 \left (99 \,\mathrm {I}+\sqrt {83949}\right )}, z \left (t \right )={\mathrm e}^{\frac {\left (\left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {2}{3}}+3 \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}+150\right ) t}{3 \left (-594+6 \,\mathrm {I} \sqrt {3}\, \sqrt {27983}\right )^{\frac {1}{3}}}} c_{1} +{\mathrm e}^{-\frac {5 \left (\sqrt {2}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )-3 \sqrt {2}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )-\frac {3}{5}\right ) t}{3}} c_{2} +c_{3} {\mathrm e}^{-\frac {5 \left (\sqrt {2}\, \sqrt {3}\, \sin \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )+3 \sqrt {2}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {83949}}{99}\right )}{3}+\frac {\pi }{6}\right )-\frac {3}{5}\right ) t}{3}}\right \} \end {array} \]
✗ Solution by Maple
dsolve([diff(x(t),t)=-3*x(t)+4*y(t)+exp(-t)*sin(2*t),diff(y(t),t)=5*x(t)+9*z(t)+4*exp(-t)*cos(2*t),diff(z(t),t)=y(t)+6*z(t)-exp(-t)],singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.649 (sec). Leaf size: 2949
DSolve[{x'[t]==-3*x[t]+4*y[t]+Exp[-t]*Sin[2*t],y'[t]==5*x[t]+9*z[t]+4*Exp[-t]*Cos[2*t],z'[t]==y[t]+6*z[t]-Exp[-t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
Too large to display