problem section 10.6, problem 5

Internal problem ID [1640]
Internal ﬁle name [OUTPUT/1641_Sunday_June_05_2022_02_25_27_AM_40857027/index.tex]

Book: Elementary diﬀerential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section: Chapter 10 Linear system of Diﬀerential equations. Section 10.6, constant coeﬃcient homogeneous system III. Page 566
Problem number: section 10.6, problem 5.
ODE order: 1.
ODE degree: 1.

Solve \begin {align*} y_{1}^{\prime }\left (t \right )&=-3 y_{1} \left (t \right )-3 y_{2} \left (t \right )+y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=2 y_{2} \left (t \right )+2 y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=5 y_{1} \left (t \right )+y_{2} \left (t \right )+y_{3} \left (t \right ) \end {align*}

23.5.1 Solution using Matrix exponential method

In this method, we will assume we have found the matrix exponential \(e^{A t}\) allready. There are diﬀerent methods to determine this but will not be shown here. This is a system of linear ODE’s given as \begin {align*} \vec {x}'(t) &= A\, \vec {x}(t) \end {align*}

Or \begin {align*} \left [\begin {array}{c} y_{1}^{\prime }\left (t \right ) \\ y_{2}^{\prime }\left (t \right ) \\ y_{3}^{\prime }\left (t \right ) \end {array}\right ] &= \left [\begin {array}{ccc} -3 & -3 & 1 \\ 0 & 2 & 2 \\ 5 & 1 & 1 \end {array}\right ]\, \left [\begin {array}{c} y_{1} \left (t \right ) \\ y_{2} \left (t \right ) \\ y_{3} \left (t \right ) \end {array}\right ] \end {align*}

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

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

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

23.5.2 Solution using explicit Eigenvalue and Eigenvector method

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

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

Expanding gives \begin {align*} \operatorname {det} \left (\left [\begin {array}{ccc} -3 & -3 & 1 \\ 0 & 2 & 2 \\ 5 & 1 & 1 \end {array}\right ]-\lambda \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) &= 0 \end {align*}

Therefore \begin {align*} \operatorname {det} \left (\left [\begin {array}{ccc} -3-\lambda & -3 & 1 \\ 0 & 2-\lambda & 2 \\ 5 & 1 & 1-\lambda \end {array}\right ]\right ) &= 0 \end {align*}

Which gives the characteristic equation \begin {align*} \lambda ^{3}-14 \lambda +40&=0 \end {align*}

The roots of the above are the eigenvalues. \begin {align*} \lambda _1 &= -\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\\ \lambda _2 &= \frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}\\ \lambda _3 &= \frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}


eigenvalue	algebraic multiplicity	type of eigenvalue

\(\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}\)	\(1\)	complex eigenvalue

\(\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}\)	\(1\)	complex eigenvalue

\(-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\)	\(1\)	real eigenvalue

Considering the eigenvalue \(\lambda _{1} = -\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\)

We need to solve \(A \vec {v} = \lambda \vec {v}\) or \((A-\lambda I) \vec {v} = \vec {0}\) which becomes \begin {align*} \left (\left [\begin {array}{ccc} -3 & -3 & 1 \\ 0 & 2 & 2 \\ 5 & 1 & 1 \end {array}\right ] - \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccc} \frac {\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-9 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+42}{3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}} & -3 & 1 \\ 0 & \frac {\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+42}{3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}} & 2 \\ 5 & 1 & \frac {\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+42}{3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}} \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \end {align*}

\begin {align*} R_{3} = R_{3}-\frac {3 \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+36 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+42\right ) \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}} R_{2}}{\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-9 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+42\right ) \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+42\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-9 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+42}{3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}&-3&1&0\\ 0&\frac {\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+42}{3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}&2&0\\ 0&0&0&0 \end {array} \right ] \end {align*}

Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} \frac {\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-9 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+42}{3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}} & -3 & 1 \\ 0 & \frac {\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+42}{3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}} & 2 \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are \(\{v_{3}\}\) and the leading variables are \(\{v_{1}, v_{2}\}\). Let \(v_{3} = t\). Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation \(\left \{v_{1} = -\frac {3 t \left (4 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+\sqrt {6042}+7 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+90\right )}{5 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+\sqrt {6042}\, \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+69 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-3 \sqrt {6042}+24}, v_{2} = -\frac {6 t \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+42}\right \}\)

Hence the solution is \[ \left [\begin {array}{c} -\frac {3 t \left (4 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+\sqrt {6042}+7 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+90\right )}{5 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+\sqrt {6042}\, \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+69 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-3 \sqrt {6042}+24} \\ -\frac {6 t \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+42} \\ t \end {array}\right ] = \left [\begin {array}{c} -\frac {3 t \left (4 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+\sqrt {6042}+7 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+90\right )}{5 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+\sqrt {6042}\, \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+69 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-3 \sqrt {6042}+24} \\ -\frac {6 t \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+42} \\ t \end {array}\right ] \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \left [\begin {array}{c} -\frac {3 t \left (4 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+\sqrt {6042}+7 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+90\right )}{5 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+\sqrt {6042}\, \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+69 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-3 \sqrt {6042}+24} \\ -\frac {6 t \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+42} \\ t \end {array}\right ] = t \left [\begin {array}{c} -\frac {3 \left (4 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+\sqrt {6042}+7 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+90\right )}{5 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+\sqrt {6042}\, \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+69 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-3 \sqrt {6042}+24} \\ -\frac {6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+42} \\ 1 \end {array}\right ] \] Let \(t = 1\) the eigenvector becomes \[ \left [\begin {array}{c} -\frac {3 t \left (4 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+\sqrt {6042}+7 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+90\right )}{5 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+\sqrt {6042}\, \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+69 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-3 \sqrt {6042}+24} \\ -\frac {6 t \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+42} \\ t \end {array}\right ] = \left [\begin {array}{c} -\frac {3 \left (4 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+\sqrt {6042}+7 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+90\right )}{5 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+\sqrt {6042}\, \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+69 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-3 \sqrt {6042}+24} \\ -\frac {6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+42} \\ 1 \end {array}\right ] \] Which is normalized to \[ \left [\begin {array}{c} -\frac {3 t \left (4 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+\sqrt {6042}+7 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+90\right )}{5 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+\sqrt {6042}\, \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+69 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-3 \sqrt {6042}+24} \\ -\frac {6 t \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+42} \\ t \end {array}\right ] = \left [\begin {array}{c} -\frac {3 \left (4 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+\sqrt {6042}+7 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+90\right )}{5 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+\sqrt {6042}\, \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+69 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-3 \sqrt {6042}+24} \\ -\frac {6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+42} \\ 1 \end {array}\right ] \] Considering the eigenvalue \(\lambda _{2} = \frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}\)

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 & -3 & 1 \\ 0 & 2 & 2 \\ 5 & 1 & 1 \end {array}\right ] - \left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}\right ) \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccc} -\frac {42+18 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+i \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) \sqrt {3}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}} & -3 & 1 \\ 0 & \frac {-42+12 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-i \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) \sqrt {3}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}} & 2 \\ 5 & 1 & \frac {-42+6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-i \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) \sqrt {3}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\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 \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -3-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}-\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}&-3&1&0\\ 0&2-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}-\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}&2&0\\ 5&1&1-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}-\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}&0 \end {array} \right ] \] \begin {align*} R_{3} = R_{3}-\frac {5 R_{1}}{-3-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}-\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {42+18 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+i \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) \sqrt {3}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}&-3&1&0\\ 0&\frac {-42+12 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-i \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) \sqrt {3}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}&2&0\\ 0&\frac {i \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}} \sqrt {3}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42 i \sqrt {3}-72 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+42}{i \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}} \sqrt {3}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42 i \sqrt {3}+18 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+42}&\frac {2 \left (48-48 i \sqrt {3}-3 i \sqrt {2014}+\sqrt {6042}\right ) \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-492-492 i \sqrt {3}-36 i \sqrt {2014}-12 \sqrt {6042}-8 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}} \left (42+18 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+i \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) \sqrt {3}+\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}\right )}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}-\frac {6 \left (i \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}} \sqrt {3}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42 i \sqrt {3}-72 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+42\right ) \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} R_{2}}{\left (i \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}} \sqrt {3}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42 i \sqrt {3}+18 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+42\right ) \left (-42+12 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-i \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) \sqrt {3}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -\frac {42+18 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+i \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) \sqrt {3}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}&-3&1&0\\ 0&\frac {-42+12 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-i \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) \sqrt {3}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}&2&0\\ 0&0&0&0 \end {array} \right ] \end {align*}

Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} -\frac {42+18 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+i \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) \sqrt {3}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}} & -3 & 1 \\ 0 & \frac {-42+12 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-i \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) \sqrt {3}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}} & 2 \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are \(\{v_{3}\}\) and the leading variables are \(\{v_{1}, v_{2}\}\). Let \(v_{3} = t\). Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation \(\left \{v_{1} = \frac {3 t \left (3 i \sqrt {2014}-7 i \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+90 i \sqrt {3}+\sqrt {3}\, \sqrt {2014}-8 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}+7 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+90\right )}{3 i \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {2014}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {2014}+69 i \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+9 i \sqrt {2014}+3 \sqrt {3}\, \sqrt {2014}+10 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-24 i \sqrt {3}-69 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-24}, v_{2} = \frac {12 t \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}{i \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}} \sqrt {3}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42 i \sqrt {3}-12 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+42}\right \}\)

Hence the solution is \[ \left [\begin {array}{c} \frac {3 t \left (3 \,\operatorname {I} \sqrt {2014}-7 \,\operatorname {I} \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+90 \,\operatorname {I} \sqrt {3}+\sqrt {3}\, \sqrt {2014}-8 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}+7 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+90\right )}{3 \,\operatorname {I} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {2014}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {2014}+69 \,\operatorname {I} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}+9 \,\operatorname {I} \sqrt {2014}+3 \sqrt {3}\, \sqrt {2014}+10 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-24 \,\operatorname {I} \sqrt {3}-69 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-24} \\ \frac {12 t \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}{\operatorname {I} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}} \sqrt {3}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42 \,\operatorname {I} \sqrt {3}-12 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+42} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {3 t \left (3 i \sqrt {2014}-7 i \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+90 i \sqrt {3}+\sqrt {3}\, \sqrt {2014}-8 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}+7 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+90\right )}{3 i \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {2014}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {2014}+69 i \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+9 i \sqrt {2014}+3 \sqrt {3}\, \sqrt {2014}+10 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-24 i \sqrt {3}-69 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-24} \\ \frac {12 t \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}{i \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}} \sqrt {3}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42 i \sqrt {3}-12 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+42} \\ t \end {array}\right ] \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \left [\begin {array}{c} \frac {3 t \left (3 \,\operatorname {I} \sqrt {2014}-7 \,\operatorname {I} \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+90 \,\operatorname {I} \sqrt {3}+\sqrt {3}\, \sqrt {2014}-8 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}+7 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+90\right )}{3 \,\operatorname {I} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {2014}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {2014}+69 \,\operatorname {I} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}+9 \,\operatorname {I} \sqrt {2014}+3 \sqrt {3}\, \sqrt {2014}+10 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-24 \,\operatorname {I} \sqrt {3}-69 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-24} \\ \frac {12 t \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}{\operatorname {I} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}} \sqrt {3}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42 \,\operatorname {I} \sqrt {3}-12 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+42} \\ t \end {array}\right ] = t \left [\begin {array}{c} \frac {9 i \sqrt {2014}-21 i \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+270 i \sqrt {3}+3 \sqrt {3}\, \sqrt {2014}-24 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}+21 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+270}{3 i \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {2014}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {2014}+69 i \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+9 i \sqrt {2014}+3 \sqrt {3}\, \sqrt {2014}+10 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-24 i \sqrt {3}-69 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-24} \\ \frac {12 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}{i \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}} \sqrt {3}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42 i \sqrt {3}-12 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+42} \\ 1 \end {array}\right ] \] Let \(t = 1\) the eigenvector becomes \[ \left [\begin {array}{c} \frac {3 t \left (3 \,\operatorname {I} \sqrt {2014}-7 \,\operatorname {I} \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+90 \,\operatorname {I} \sqrt {3}+\sqrt {3}\, \sqrt {2014}-8 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}+7 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+90\right )}{3 \,\operatorname {I} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {2014}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {2014}+69 \,\operatorname {I} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}+9 \,\operatorname {I} \sqrt {2014}+3 \sqrt {3}\, \sqrt {2014}+10 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-24 \,\operatorname {I} \sqrt {3}-69 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-24} \\ \frac {12 t \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}{\operatorname {I} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}} \sqrt {3}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42 \,\operatorname {I} \sqrt {3}-12 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+42} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {9 i \sqrt {2014}-21 i \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+270 i \sqrt {3}+3 \sqrt {3}\, \sqrt {2014}-24 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}+21 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+270}{3 i \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {2014}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {2014}+69 i \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+9 i \sqrt {2014}+3 \sqrt {3}\, \sqrt {2014}+10 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-24 i \sqrt {3}-69 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-24} \\ \frac {12 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}{i \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}} \sqrt {3}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42 i \sqrt {3}-12 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+42} \\ 1 \end {array}\right ] \] Which is normalized to \[ \left [\begin {array}{c} \frac {3 t \left (3 \,\operatorname {I} \sqrt {2014}-7 \,\operatorname {I} \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+90 \,\operatorname {I} \sqrt {3}+\sqrt {3}\, \sqrt {2014}-8 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}+7 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+90\right )}{3 \,\operatorname {I} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {2014}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {2014}+69 \,\operatorname {I} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}+9 \,\operatorname {I} \sqrt {2014}+3 \sqrt {3}\, \sqrt {2014}+10 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-24 \,\operatorname {I} \sqrt {3}-69 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-24} \\ \frac {12 t \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}{\operatorname {I} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}} \sqrt {3}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42 \,\operatorname {I} \sqrt {3}-12 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+42} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {9 i \sqrt {2014}-21 i \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+270 i \sqrt {3}+3 \sqrt {3}\, \sqrt {2014}-24 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}+21 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+270}{3 i \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {2014}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {2014}+69 i \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+9 i \sqrt {2014}+3 \sqrt {3}\, \sqrt {2014}+10 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-24 i \sqrt {3}-69 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-24} \\ \frac {12 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}{i \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}} \sqrt {3}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42 i \sqrt {3}-12 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+42} \\ 1 \end {array}\right ] \] Considering the eigenvalue \(\lambda _{3} = \frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}\)

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 & -3 & 1 \\ 0 & 2 & 2 \\ 5 & 1 & 1 \end {array}\right ] - \left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}\right ) \left [\begin {array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccc} \frac {-42-18 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+i \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) \sqrt {3}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}} & -3 & 1 \\ 0 & \frac {-42+12 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+i \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) \sqrt {3}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}} & 2 \\ 5 & 1 & \frac {-42+6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+i \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) \sqrt {3}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\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 \[ \left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} -3-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}-\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}&-3&1&0\\ 0&2-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}-\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}&2&0\\ 5&1&1-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}-\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}&0 \end {array} \right ] \] \begin {align*} R_{3} = R_{3}-\frac {5 R_{1}}{-3-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}-\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {-42-18 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+i \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) \sqrt {3}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}&-3&1&0\\ 0&\frac {-42+12 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+i \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) \sqrt {3}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}&2&0\\ 0&\frac {i \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sqrt {3}-\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42 i \sqrt {3}+72 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-42}{i \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sqrt {3}-\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42 i \sqrt {3}-18 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-42}&\frac {-2 \left (48 i \sqrt {3}+3 i \sqrt {2014}+\sqrt {6042}+48\right ) \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+492-492 i \sqrt {3}-36 i \sqrt {2014}+12 \sqrt {6042}+8 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}}{\left (-42-18 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+i \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) \sqrt {3}-\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}\right ) \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}&0 \end {array} \right ] \end {align*}

\begin {align*} R_{3} = R_{3}-\frac {6 \left (i \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sqrt {3}-\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42 i \sqrt {3}+72 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-42\right ) \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} R_{2}}{\left (i \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sqrt {3}-\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42 i \sqrt {3}-18 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-42\right ) \left (-42+12 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+i \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) \sqrt {3}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} \frac {-42-18 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+i \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) \sqrt {3}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}&-3&1&0\\ 0&\frac {-42+12 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+i \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) \sqrt {3}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}&2&0\\ 0&0&0&0 \end {array} \right ] \end {align*}

Therefore the system in Echelon form is \[ \left [\begin {array}{ccc} \frac {-42-18 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+i \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) \sqrt {3}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}} & -3 & 1 \\ 0 & \frac {-42+12 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+i \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) \sqrt {3}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}} & 2 \\ 0 & 0 & 0 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are \(\{v_{3}\}\) and the leading variables are \(\{v_{1}, v_{2}\}\). Let \(v_{3} = t\). Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation \(\left \{v_{1} = \frac {3 t \left (3 i \sqrt {2014}-7 i \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+90 i \sqrt {3}-\sqrt {3}\, \sqrt {2014}+8 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-7 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-90\right )}{3 i \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {2014}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {2014}+69 i \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+9 i \sqrt {2014}-3 \sqrt {3}\, \sqrt {2014}-10 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-24 i \sqrt {3}+69 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+24}, v_{2} = -\frac {12 t \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{i \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sqrt {3}-\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42 i \sqrt {3}+12 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-42}\right \}\)

Hence the solution is \[ \left [\begin {array}{c} \frac {3 t \left (3 \,\operatorname {I} \sqrt {2014}-7 \,\operatorname {I} \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+90 \,\operatorname {I} \sqrt {3}-\sqrt {3}\, \sqrt {2014}+8 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-7 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-90\right )}{3 \,\operatorname {I} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {2014}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {2014}+69 \,\operatorname {I} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}+9 \,\operatorname {I} \sqrt {2014}-3 \sqrt {3}\, \sqrt {2014}-10 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-24 \,\operatorname {I} \sqrt {3}+69 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+24} \\ -\frac {12 t \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{\operatorname {I} \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sqrt {3}-\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42 \,\operatorname {I} \sqrt {3}+12 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-42} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {3 t \left (3 i \sqrt {2014}-7 i \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+90 i \sqrt {3}-\sqrt {3}\, \sqrt {2014}+8 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-7 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-90\right )}{3 i \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {2014}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {2014}+69 i \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+9 i \sqrt {2014}-3 \sqrt {3}\, \sqrt {2014}-10 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-24 i \sqrt {3}+69 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+24} \\ -\frac {12 t \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{i \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sqrt {3}-\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42 i \sqrt {3}+12 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-42} \\ t \end {array}\right ] \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \left [\begin {array}{c} \frac {3 t \left (3 \,\operatorname {I} \sqrt {2014}-7 \,\operatorname {I} \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+90 \,\operatorname {I} \sqrt {3}-\sqrt {3}\, \sqrt {2014}+8 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-7 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-90\right )}{3 \,\operatorname {I} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {2014}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {2014}+69 \,\operatorname {I} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}+9 \,\operatorname {I} \sqrt {2014}-3 \sqrt {3}\, \sqrt {2014}-10 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-24 \,\operatorname {I} \sqrt {3}+69 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+24} \\ -\frac {12 t \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{\operatorname {I} \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sqrt {3}-\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42 \,\operatorname {I} \sqrt {3}+12 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-42} \\ t \end {array}\right ] = t \left [\begin {array}{c} \frac {9 i \sqrt {2014}-21 i \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+270 i \sqrt {3}-3 \sqrt {3}\, \sqrt {2014}+24 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-21 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-270}{3 i \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {2014}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {2014}+69 i \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+9 i \sqrt {2014}-3 \sqrt {3}\, \sqrt {2014}-10 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-24 i \sqrt {3}+69 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+24} \\ -\frac {12 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{i \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sqrt {3}-\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42 i \sqrt {3}+12 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-42} \\ 1 \end {array}\right ] \] Let \(t = 1\) the eigenvector becomes \[ \left [\begin {array}{c} \frac {3 t \left (3 \,\operatorname {I} \sqrt {2014}-7 \,\operatorname {I} \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+90 \,\operatorname {I} \sqrt {3}-\sqrt {3}\, \sqrt {2014}+8 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-7 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-90\right )}{3 \,\operatorname {I} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {2014}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {2014}+69 \,\operatorname {I} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}+9 \,\operatorname {I} \sqrt {2014}-3 \sqrt {3}\, \sqrt {2014}-10 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-24 \,\operatorname {I} \sqrt {3}+69 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+24} \\ -\frac {12 t \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{\operatorname {I} \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sqrt {3}-\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42 \,\operatorname {I} \sqrt {3}+12 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-42} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {9 i \sqrt {2014}-21 i \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+270 i \sqrt {3}-3 \sqrt {3}\, \sqrt {2014}+24 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-21 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-270}{3 i \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {2014}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {2014}+69 i \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+9 i \sqrt {2014}-3 \sqrt {3}\, \sqrt {2014}-10 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-24 i \sqrt {3}+69 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+24} \\ -\frac {12 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{i \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sqrt {3}-\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42 i \sqrt {3}+12 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-42} \\ 1 \end {array}\right ] \] Which is normalized to \[ \left [\begin {array}{c} \frac {3 t \left (3 \,\operatorname {I} \sqrt {2014}-7 \,\operatorname {I} \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+90 \,\operatorname {I} \sqrt {3}-\sqrt {3}\, \sqrt {2014}+8 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-7 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-90\right )}{3 \,\operatorname {I} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {2014}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {2014}+69 \,\operatorname {I} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}+9 \,\operatorname {I} \sqrt {2014}-3 \sqrt {3}\, \sqrt {2014}-10 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-24 \,\operatorname {I} \sqrt {3}+69 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+24} \\ -\frac {12 t \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{\operatorname {I} \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sqrt {3}-\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42 \,\operatorname {I} \sqrt {3}+12 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-42} \\ t \end {array}\right ] = \left [\begin {array}{c} \frac {9 i \sqrt {2014}-21 i \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+270 i \sqrt {3}-3 \sqrt {3}\, \sqrt {2014}+24 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-21 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-270}{3 i \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {2014}+\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {2014}+69 i \sqrt {3}\, \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+9 i \sqrt {2014}-3 \sqrt {3}\, \sqrt {2014}-10 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-24 i \sqrt {3}+69 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+24} \\ -\frac {12 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{i \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sqrt {3}-\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42 i \sqrt {3}+12 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-42} \\ 1 \end {array}\right ] \] The following table gives a summary of this result. It shows for each eigenvalue the algebraic multiplicity \(m\), and its geometric multiplicity \(k\) and the eigenvectors associated with the eigenvalue. If \(m>k\) then the eigenvalue is defective which means the number of normal linearly independent eigenvectors associated with this eigenvalue (called the geometric multiplicity \(k\)) does not equal the algebraic multiplicity \(m\), and we need to determine an additional \(m-k\) generalized eigenvectors for this eigenvalue.


	multiplicity

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

\(-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\)	\(1\)	\(1\)	No	\(\left [\begin {array}{c} \frac {-8-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}}{\left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-2\right ) \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+3\right )} \\ \frac {2}{-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-2} \\ 1 \end {array}\right ]\)

\(\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}\)	\(1\)	\(1\)	No	\(\left [\begin {array}{c} \frac {-8+\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}}{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}-2\right ) \left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}+3\right )} \\ \frac {2}{\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}-2} \\ 1 \end {array}\right ]\)

\(\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}\)	\(1\)	\(1\)	No	\(\left [\begin {array}{c} \frac {-8+\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}}{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}-2\right ) \left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}+3\right )} \\ \frac {2}{\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}-2} \\ 1 \end {array}\right ]\)

Now that we found the eigenvalues and associated eigenvectors, we will go over each eigenvalue and generate the solution basis. The only problem we need to take care of is if the eigenvalue is defective. Since eigenvalue \(-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\) is real and distinct then the corresponding eigenvector solution is \begin {align*} \vec {x}_{1}(t) &= \vec {v}_{1} e^{\left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t}\\ &= \left [\begin {array}{c} \frac {-8-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}}{\left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-2\right ) \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+3\right )} \\ \frac {2}{-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-2} \\ 1 \end {array}\right ] e^{\left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t} \end {align*}

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

Which is written as \begin {align*} \left [\begin {array}{c} y_{1} \left (t \right ) \\ y_{2} \left (t \right ) \\ y_{3} \left (t \right ) \end {array}\right ] &= c_{1} \left [\begin {array}{c} \frac {{\mathrm e}^{\left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t} \left (-8-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{\left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-2\right ) \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+3\right )} \\ \frac {2 \,{\mathrm e}^{\left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t}}{-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-2} \\ {\mathrm e}^{\left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t} \end {array}\right ] + c_{2} \left [\begin {array}{c} \frac {{\mathrm e}^{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} \left (-8+\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}-2\right ) \left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}+3\right )} \\ \frac {2 \,{\mathrm e}^{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}}{\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}-2} \\ {\mathrm e}^{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} \end {array}\right ] + c_{3} \left [\begin {array}{c} \frac {{\mathrm e}^{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} \left (-8+\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}-2\right ) \left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}+3\right )} \\ \frac {2 \,{\mathrm e}^{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}}{\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}-2} \\ {\mathrm e}^{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} \end {array}\right ] \end {align*}

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

23.5.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y_{1}^{\prime }\left (t \right )=-3 y_{1} \left (t \right )-3 y_{2} \left (t \right )+y_{3} \left (t \right ), y_{2}^{\prime }\left (t \right )=2 y_{2} \left (t \right )+2 y_{3} \left (t \right ), y_{3}^{\prime }\left (t \right )=5 y_{1} \left (t \right )+y_{2} \left (t \right )+y_{3} \left (t \right )\right ] \\ \bullet & {} & \textrm {Define vector}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{\textit {y\_\_}}}\left (t \right )=\left [\begin {array}{c} y_{1} \left (t \right ) \\ y_{2} \left (t \right ) \\ y_{3} \left (t \right ) \end {array}\right ] \\ \bullet & {} & \textrm {Convert system into a vector equation}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{\textit {y\_\_}}}^{\prime }\left (t \right )=\left [\begin {array}{ccc} -3 & -3 & 1 \\ 0 & 2 & 2 \\ 5 & 1 & 1 \end {array}\right ]\cdot {\moverset {\rightarrow }{\textit {y\_\_}}}\left (t \right )+\left [\begin {array}{c} 0 \\ 0 \\ 0 \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{\textit {y\_\_}}}^{\prime }\left (t \right )=\left [\begin {array}{ccc} -3 & -3 & 1 \\ 0 & 2 & 2 \\ 5 & 1 & 1 \end {array}\right ]\cdot {\moverset {\rightarrow }{\textit {y\_\_}}}\left (t \right ) \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{ccc} -3 & -3 & 1 \\ 0 & 2 & 2 \\ 5 & 1 & 1 \end {array}\right ] \\ \bullet & {} & \textrm {Rewrite the system as}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{\textit {y\_\_}}}^{\prime }\left (t \right )=A \cdot {\moverset {\rightarrow }{\textit {y\_\_}}}\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 (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}, \left [\begin {array}{c} \frac {-8-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}}{\left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-2\right ) \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+3\right )} \\ \frac {2}{-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-2} \\ 1 \end {array}\right ]\right ], \left [\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {-8+\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}}{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}-2\right ) \left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}+3\right )} \\ \frac {2}{\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}-2} \\ 1 \end {array}\right ]\right ], \left [\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {-8+\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}}{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}-2\right ) \left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}+3\right )} \\ \frac {2}{\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}-2} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}, \left [\begin {array}{c} \frac {-8-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}}{\left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-2\right ) \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+3\right )} \\ \frac {2}{-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-2} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{\textit {y\_\_}}}_{1}={\mathrm e}^{\left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t}\cdot \left [\begin {array}{c} \frac {-8-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}}{\left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-2\right ) \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+3\right )} \\ \frac {2}{-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {-8+\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}}{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}-2\right ) \left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}+3\right )} \\ \frac {2}{\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}-2} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}\cdot \left [\begin {array}{c} \frac {-8+\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}}{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}-2\right ) \left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}+3\right )} \\ \frac {2}{\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}-2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Use Euler identity to write solution in terms of}\hspace {3pt} \sin \hspace {3pt}\textrm {and}\hspace {3pt} \cos \\ {} & {} & {\mathrm e}^{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t}\cdot \left (\cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t}{2}\right )\right )\cdot \left [\begin {array}{c} \frac {-8+\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}}{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}-2\right ) \left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}+3\right )} \\ \frac {2}{\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}-2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t}\cdot \left [\begin {array}{c} \frac {\left (\cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t}{2}\right )\right ) \left (-8+\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}\right )}{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}-2\right ) \left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}+3\right )} \\ \frac {2 \left (\cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t}{2}\right )\right )}{\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )}{2}-2} \\ \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t}{2}\right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{\textit {y\_\_}}}_{2}\left (t \right )={\mathrm e}^{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t}\cdot \left [\begin {array}{c} -\frac {9 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}} \left (4 \left (540+6 \sqrt {6042}\right )^{\frac {5}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-5 \left (540+6 \sqrt {6042}\right )^{\frac {5}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+17 \left (540+6 \sqrt {6042}\right )^{\frac {4}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+21 \left (540+6 \sqrt {6042}\right )^{\frac {4}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-714 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-72504 \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+882 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-7056 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-1080 \sqrt {3}\, \sqrt {6042}\, \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+55080 \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-8820 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+612 \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) \sqrt {6042}\right )}{\left (\left (540+6 \sqrt {6042}\right )^{\frac {4}{3}}-6 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-252 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-36 \sqrt {6042}-1476\right ) \left (\left (540+6 \sqrt {6042}\right )^{\frac {4}{3}}+39 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+378 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+54 \sqrt {6042}+6624\right )} \\ \frac {3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}} \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-12 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-42 \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+42 \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )\right )}{\left (540+6 \sqrt {6042}\right )^{\frac {4}{3}}-6 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-252 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-36 \sqrt {6042}-1476} \\ \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t}{2}\right ) \end {array}\right ], {\moverset {\rightarrow }{\textit {y\_\_}}}_{3}\left (t \right )={\mathrm e}^{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t}\cdot \left [\begin {array}{c} \frac {9 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}} \left (4 \left (540+6 \sqrt {6042}\right )^{\frac {5}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+5 \left (540+6 \sqrt {6042}\right )^{\frac {5}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+17 \left (540+6 \sqrt {6042}\right )^{\frac {4}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-21 \left (540+6 \sqrt {6042}\right )^{\frac {4}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-714 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-72504 \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) \sqrt {3}-882 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-7056 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-1080 \sqrt {3}\, \sqrt {6042}\, \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-55080 \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+8820 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-612 \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) \sqrt {6042}\right )}{\left (\left (540+6 \sqrt {6042}\right )^{\frac {4}{3}}-6 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-252 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-36 \sqrt {6042}-1476\right ) \left (\left (540+6 \sqrt {6042}\right )^{\frac {4}{3}}+39 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+378 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+54 \sqrt {6042}+6624\right )} \\ -\frac {3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}} \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+12 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-42 \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) \sqrt {3}-42 \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )\right )}{\left (540+6 \sqrt {6042}\right )^{\frac {4}{3}}-6 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-252 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-36 \sqrt {6042}-1476} \\ -\sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t}{2}\right ) \end {array}\right ]\right ] \\ \bullet & {} & \textrm {General solution to the system of ODEs}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{\textit {y\_\_}}}=c_{1} {\moverset {\rightarrow }{\textit {y\_\_}}}_{1}+c_{2} {\moverset {\rightarrow }{\textit {y\_\_}}}_{2}\left (t \right )+c_{3} {\moverset {\rightarrow }{\textit {y\_\_}}}_{3}\left (t \right ) \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{\textit {y\_\_}}}=c_{1} {\mathrm e}^{\left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t}\cdot \left [\begin {array}{c} \frac {-8-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}}{\left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-2\right ) \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}+3\right )} \\ \frac {2}{-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}-\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}-2} \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t}\cdot \left [\begin {array}{c} -\frac {9 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}} \left (4 \left (540+6 \sqrt {6042}\right )^{\frac {5}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-5 \left (540+6 \sqrt {6042}\right )^{\frac {5}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+17 \left (540+6 \sqrt {6042}\right )^{\frac {4}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+21 \left (540+6 \sqrt {6042}\right )^{\frac {4}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-714 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-72504 \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+882 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-7056 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-1080 \sqrt {3}\, \sqrt {6042}\, \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+55080 \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-8820 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+612 \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) \sqrt {6042}\right )}{\left (\left (540+6 \sqrt {6042}\right )^{\frac {4}{3}}-6 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-252 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-36 \sqrt {6042}-1476\right ) \left (\left (540+6 \sqrt {6042}\right )^{\frac {4}{3}}+39 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+378 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+54 \sqrt {6042}+6624\right )} \\ \frac {3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}} \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-12 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-42 \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+42 \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )\right )}{\left (540+6 \sqrt {6042}\right )^{\frac {4}{3}}-6 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-252 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-36 \sqrt {6042}-1476} \\ \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t}{2}\right ) \end {array}\right ]+c_{3} {\mathrm e}^{\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{6}+\frac {7}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t}\cdot \left [\begin {array}{c} \frac {9 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}} \left (4 \left (540+6 \sqrt {6042}\right )^{\frac {5}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+5 \left (540+6 \sqrt {6042}\right )^{\frac {5}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+17 \left (540+6 \sqrt {6042}\right )^{\frac {4}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-21 \left (540+6 \sqrt {6042}\right )^{\frac {4}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-714 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-72504 \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) \sqrt {3}-882 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-7056 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-1080 \sqrt {3}\, \sqrt {6042}\, \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-55080 \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+8820 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-612 \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) \sqrt {6042}\right )}{\left (\left (540+6 \sqrt {6042}\right )^{\frac {4}{3}}-6 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-252 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-36 \sqrt {6042}-1476\right ) \left (\left (540+6 \sqrt {6042}\right )^{\frac {4}{3}}+39 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+378 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}+54 \sqrt {6042}+6624\right )} \\ -\frac {3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}} \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+12 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-42 \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) \sqrt {3}-42 \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )\right )}{\left (540+6 \sqrt {6042}\right )^{\frac {4}{3}}-6 \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-252 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-36 \sqrt {6042}-1476} \\ -\sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{3}+\frac {14}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) t}{2}\right ) \end {array}\right ] \\ \bullet & {} & \textrm {Substitute in vector of dependent variables}\hspace {3pt} \\ {} & {} & \left [\begin {array}{c} y_{1} \left (t \right ) \\ y_{2} \left (t \right ) \\ y_{3} \left (t \right ) \end {array}\right ]=\left [\begin {array}{c} \frac {1031 \left ({\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{2 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \left (\left (\left (\sqrt {3}\, c_{2} +3 c_{3} \right ) \sqrt {2014}+\frac {80442 c_{2}}{1031}+\frac {80442 c_{3} \sqrt {3}}{1031}\right ) \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+\frac {\left (12 \left (\sqrt {3}\, c_{2} -3 c_{3} \right ) \sqrt {2014}+1423 c_{2} -1423 c_{3} \sqrt {3}\right ) \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{1031}-\frac {5760 c_{2} \left (\sqrt {3}\, \sqrt {2014}+\frac {2357}{30}\right )}{1031}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}\right )-\frac {80442 \,{\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{2 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \left (\left (\frac {1031 \left (-\frac {c_{3} \sqrt {3}}{3}+c_{2} \right ) \sqrt {2014}}{26814}+\sqrt {3}\, c_{2} -c_{3} \right ) \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+\frac {\left (2 \left (-\frac {c_{3} \sqrt {3}}{3}-c_{2} \right ) \sqrt {2014}-\frac {1423 \sqrt {3}\, c_{2}}{18}-\frac {1423 c_{3}}{18}\right ) \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{4469}+\frac {320 \left (\sqrt {3}\, \sqrt {2014}+\frac {2357}{30}\right ) c_{3}}{4469}\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}\right )}{1031}-2 \left (\frac {226272}{1031}+\left (\sqrt {3}\, \sqrt {2014}+\frac {80442}{1031}\right ) \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+\frac {\left (12 \sqrt {3}\, \sqrt {2014}+1423\right ) \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{1031}+\frac {2880 \sqrt {3}\, \sqrt {2014}}{1031}\right ) c_{1} \right ) {\mathrm e}^{-\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}}}{900 \left (30 \sqrt {3}\, \sqrt {2014}+2357\right )} \\ \frac {\left ({\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{2 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \left (\left (\left (\sqrt {3}\, c_{2} -3 c_{3} \right ) \sqrt {2014}-48 c_{3} \sqrt {3}+48 c_{2} \right ) \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+2 c_{2} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}+6 \left (-\sqrt {3}\, c_{2} -3 c_{3} \right ) \sqrt {2014}-246 c_{3} \sqrt {3}-246 c_{2} \right ) \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}\right )+48 \left (\left (\frac {\left (\frac {c_{3} \sqrt {3}}{3}+c_{2} \right ) \sqrt {2014}}{16}+\sqrt {3}\, c_{2} +c_{3} \right ) \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+\frac {\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}} c_{3}}{24}+\frac {\left (-c_{3} \sqrt {3}+3 c_{2} \right ) \sqrt {2014}}{8}+\frac {41 \sqrt {3}\, c_{2}}{8}-\frac {41 c_{3}}{8}\right ) {\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{2 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}\right )-2 \left (\left (\sqrt {3}\, \sqrt {2014}+48\right ) \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-6 \sqrt {3}\, \sqrt {2014}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-246\right ) c_{1} \right ) {\mathrm e}^{-\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}{180 \left (\sqrt {3}\, \sqrt {2014}+90\right )} \\ \left (\sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}\right ) {\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{2 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} c_{3} +{\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{2 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}\right ) c_{2} +c_{1} \right ) {\mathrm e}^{-\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \end {array}\right ] \\ \bullet & {} & \textrm {Solution to the system of ODEs}\hspace {3pt} \\ {} & {} & \left \{y_{1} \left (t \right )=\frac {1031 \left ({\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{2 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \left (\left (\left (\sqrt {3}\, c_{2} +3 c_{3} \right ) \sqrt {2014}+\frac {80442 c_{2}}{1031}+\frac {80442 c_{3} \sqrt {3}}{1031}\right ) \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+\frac {\left (12 \left (\sqrt {3}\, c_{2} -3 c_{3} \right ) \sqrt {2014}+1423 c_{2} -1423 c_{3} \sqrt {3}\right ) \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{1031}-\frac {5760 c_{2} \left (\sqrt {3}\, \sqrt {2014}+\frac {2357}{30}\right )}{1031}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}\right )-\frac {80442 \,{\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{2 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \left (\left (\frac {1031 \left (-\frac {c_{3} \sqrt {3}}{3}+c_{2} \right ) \sqrt {2014}}{26814}+\sqrt {3}\, c_{2} -c_{3} \right ) \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+\frac {\left (2 \left (-\frac {c_{3} \sqrt {3}}{3}-c_{2} \right ) \sqrt {2014}-\frac {1423 \sqrt {3}\, c_{2}}{18}-\frac {1423 c_{3}}{18}\right ) \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{4469}+\frac {320 \left (\sqrt {3}\, \sqrt {2014}+\frac {2357}{30}\right ) c_{3}}{4469}\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}\right )}{1031}-2 \left (\frac {226272}{1031}+\left (\sqrt {3}\, \sqrt {2014}+\frac {80442}{1031}\right ) \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+\frac {\left (12 \sqrt {3}\, \sqrt {2014}+1423\right ) \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}}{1031}+\frac {2880 \sqrt {3}\, \sqrt {2014}}{1031}\right ) c_{1} \right ) {\mathrm e}^{-\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}}}{900 \left (30 \sqrt {3}\, \sqrt {2014}+2357\right )}, y_{2} \left (t \right )=\frac {\left ({\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{2 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \left (\left (\left (\sqrt {3}\, c_{2} -3 c_{3} \right ) \sqrt {2014}-48 c_{3} \sqrt {3}+48 c_{2} \right ) \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+2 c_{2} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}+6 \left (-\sqrt {3}\, c_{2} -3 c_{3} \right ) \sqrt {2014}-246 c_{3} \sqrt {3}-246 c_{2} \right ) \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}\right )+48 \left (\left (\frac {\left (\frac {c_{3} \sqrt {3}}{3}+c_{2} \right ) \sqrt {2014}}{16}+\sqrt {3}\, c_{2} +c_{3} \right ) \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}+\frac {\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}} c_{3}}{24}+\frac {\left (-c_{3} \sqrt {3}+3 c_{2} \right ) \sqrt {2014}}{8}+\frac {41 \sqrt {3}\, c_{2}}{8}-\frac {41 c_{3}}{8}\right ) {\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{2 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}\right )-2 \left (\left (\sqrt {3}\, \sqrt {2014}+48\right ) \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}-6 \sqrt {3}\, \sqrt {2014}-\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-246\right ) c_{1} \right ) {\mathrm e}^{-\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}{180 \left (\sqrt {3}\, \sqrt {2014}+90\right )}, y_{3} \left (t \right )=\left (\sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}\right ) {\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{2 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} c_{3} +{\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{2 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {3}\, \sqrt {2014}\right )^{\frac {1}{3}}}\right ) c_{2} +c_{1} \right ) {\mathrm e}^{-\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}}\right \} \end {array} \]

\begin{align*} \text {Expression too large to display} \\ y_{2} \left (t \right ) &= {\mathrm e}^{\frac {\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}}{6}+7\right ) t}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \sin \left (\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t \sqrt {3}\, 36^{\frac {1}{3}}}{36 \left (90+\sqrt {6042}\right )^{\frac {1}{3}}}\right ) c_{2} +{\mathrm e}^{\frac {\left (\frac {\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}}{6}+7\right ) t}{\left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \cos \left (\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t \sqrt {3}\, 36^{\frac {1}{3}}}{36 \left (90+\sqrt {6042}\right )^{\frac {1}{3}}}\right ) c_{3} +c_{1} {\mathrm e}^{-\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \\ y_{3} \left (t \right ) &= \frac {-2 c_{1} \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) {\mathrm e}^{-\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}}+c_{2} \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) {\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+c_{2} {\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \left (\sqrt {3}\, \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42 \sqrt {3}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )+c_{3} \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) {\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-c_{3} {\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \left (\sqrt {3}\, \left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42 \sqrt {3}\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right )-12 c_{1} {\mathrm e}^{-\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{3 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-12 c_{2} {\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}-12 c_{3} {\mathrm e}^{\frac {\left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}+42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (540+6 \sqrt {6042}\right )^{\frac {2}{3}}-42\right ) t}{6 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}\right ) \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}}{12 \left (540+6 \sqrt {6042}\right )^{\frac {1}{3}}} \\ \end{align*}

\begin{align*} \text {y1}(t)\to \frac {1}{4} e^{-2 t} \left ((3 c_1-c_2+c_3) e^{6 t} \cos (2 t)+(c_1-3 c_2-c_3) e^{6 t} \sin (2 t)+c_1+c_2-c_3\right ) \\ \text {y2}(t)\to \frac {1}{4} e^{-2 t} \left (-(c_1-3 c_2-c_3) e^{6 t} \cos (2 t)+(3 c_1-c_2+c_3) e^{6 t} \sin (2 t)+c_1+c_2-c_3\right ) \\ \text {y3}(t)\to \frac {1}{2} e^{-2 t} \left ((c_1+c_2+c_3) e^{6 t} \cos (2 t)+2 (c_1-c_2) e^{6 t} \sin (2 t)-c_1-c_2+c_3\right ) \\ \end{align*}

23.5 problem section 10.6, problem 5

23.5.1 Solution using Matrix exponential method

23.5.2 Solution using explicit Eigenvalue and Eigenvector method

23.5.3 Maple step by step solution