problem 18

Internal problem ID [7208]
Internal ﬁle name [OUTPUT/6194_Sunday_June_05_2022_04_27_35_PM_24677674/index.tex]

Book: Own collection of miscellaneous problems
Section: section 3.0
Problem number: 18.
ODE order: 3.
ODE degree: 1.

The type(s) of ODE detected by this program : "higher_order_ODE_non_constant_coeﬃcients_of_type_Euler"

\[ \boxed {x^{4} y^{\prime \prime \prime }+x^{3} y^{\prime \prime }+y^{\prime } x^{2}+y x=0} \] This is Euler ODE of higher order. Let $y = x^{\lambda }$. Hence \begin {align*} y^{\prime } &= \lambda \,x^{\lambda -1}\\ y^{\prime \prime } &= \lambda \left (\lambda -1\right ) x^{\lambda -2}\\ y^{\prime \prime \prime } &= \lambda \left (\lambda -1\right ) \left (\lambda -2\right ) x^{\lambda -3} \end {align*}

Substituting these back into \[ y^{\prime \prime \prime } x^{3}+y^{\prime \prime } x^{2}+y^{\prime } x +y = 0 \] gives \[ x \lambda \,x^{\lambda -1}+x^{2} \lambda \left (\lambda -1\right ) x^{\lambda -2}+x^{3} \lambda \left (\lambda -1\right ) \left (\lambda -2\right ) x^{\lambda -3}+x^{\lambda } = 0 \] Which simpliﬁes to \[ \lambda \,x^{\lambda }+\lambda \left (\lambda -1\right ) x^{\lambda }+\lambda \left (\lambda -1\right ) \left (\lambda -2\right ) x^{\lambda }+x^{\lambda } = 0 \] And since $x^{\lambda }\neq 0$ then dividing through by $x^{\lambda }$, the above becomes

\[ \lambda +\lambda \left (\lambda -1\right )+\lambda \left (\lambda -1\right ) \left (\lambda -2\right )+1 = 0 \] Simplifying gives the characteristic equation as \[ \lambda ^{3}-2 \lambda ^{2}+2 \lambda +1 = 0 \] Solving the above gives the following roots \begin {align*} \lambda _1 &= -\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}\\ \lambda _2 &= \frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )}{2}\\ \lambda _3 &= \frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}


root	multiplicity	type of root

$\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3} \pm -\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )}{2} i$	$1$	complex conjugate root

$-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}$	$1$	real root

The solution is generated by going over the above table. For each real root $\lambda $ of multiplicity one generates a $c_1x^{\lambda }$ basis solution. Each real root of multiplicty two, generates $c_1x^{\lambda }$ and $c_2x^{\lambda } \ln \left (x \right )$ basis solutions. Each real root of multiplicty three, generates $c_1x^{\lambda }$ and $c_2x^{\lambda } \ln \left (x \right )$ and $c_3x^{\lambda } \ln \left (x \right )^{2}$ basis solutions, and so on. Each complex root $\alpha \pm i \beta $ of multiplicity one generates $x^{\alpha } \left (c_1\cos (\beta \ln \left (x \right ))+c_2\sin (\beta \ln \left (x \right ))\right )$ basis solutions. And each complex root $\alpha \pm i \beta $ of multiplicity two generates $\ln \left (x \right ) x^{\alpha }\left (c_1\cos (\beta \ln \left (x \right ))+c_2\sin (\beta \ln \left (x \right ))\right )$ basis solutions. And each complex root $\alpha \pm i \beta $ of multiplicity three generates $\ln \left (x \right )^{2} x^{\alpha }\left (c_1\cos (\beta \ln \left (x \right ))+c_2\sin (\beta \ln \left (x \right ))\right )$ basis solutions. And so on. Using the above show that the solution is

\[ y = x^{\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}} \left (c_{1} \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \ln \left (x \right )}{2}\right )-c_{2} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \ln \left (x \right )}{2}\right )\right )+c_{3} x^{-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= x^{\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \ln \left (x \right )}{2}\right )\\ y_2 &= -x^{\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \ln \left (x \right )}{2}\right )\\ y_3 &= x^{-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= x^{\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}} \left (c_{1} \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \ln \left (x \right )}{2}\right )-c_{2} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \ln \left (x \right )}{2}\right )\right )+c_{3} x^{-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}} \\ \end{align*}

Veriﬁcation of solutions

3.18.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y^{\prime \prime }\right ) x^{3}+\left (\frac {d}{d x}y^{\prime }\right ) x^{2}+y^{\prime } x +y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d}{d x}y^{\prime \prime } \\ \bullet & {} & \textrm {Isolate 3rd derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }=-\frac {y}{x^{3}}-\frac {\left (\frac {d}{d x}y^{\prime }\right ) x +y^{\prime }}{x^{2}} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }+\frac {\frac {d}{d x}y^{\prime }}{x}+\frac {y^{\prime }}{x^{2}}+\frac {y}{x^{3}}=0 \\ \bullet & {} & \textrm {Multiply by denominators of the ODE}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y^{\prime \prime }\right ) x^{3}+\left (\frac {d}{d x}y^{\prime }\right ) x^{2}+y^{\prime } x +y=0 \\ \bullet & {} & \textrm {Make a change of variables}\hspace {3pt} \\ {} & {} & t =\ln \left (x \right ) \\ \square & {} & \textrm {Substitute the change of variables back into the ODE}\hspace {3pt} \\ {} & \circ & \textrm {Calculate the}\hspace {3pt} \hspace {3pt}\textrm {1st}\hspace {3pt} \hspace {3pt}\textrm {derivative of}\hspace {3pt} \hspace {3pt}\textrm {y}\hspace {3pt} \hspace {3pt}\textrm {with respect to}\hspace {3pt} \hspace {3pt}\textrm {x}\hspace {3pt} \hspace {3pt}\textrm {, using the chain rule}\hspace {3pt} \\ {} & {} & y^{\prime }=\left (\frac {d}{d t}y \left (t \right )\right ) t^{\prime }\left (x \right ) \\ {} & \circ & \textrm {Compute derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\frac {d}{d t}y \left (t \right )}{x} \\ {} & \circ & \textrm {Calculate the}\hspace {3pt} \hspace {3pt}\textrm {2nd}\hspace {3pt} \hspace {3pt}\textrm {derivative of}\hspace {3pt} \hspace {3pt}\textrm {y}\hspace {3pt} \hspace {3pt}\textrm {with respect to}\hspace {3pt} \hspace {3pt}\textrm {x}\hspace {3pt} \hspace {3pt}\textrm {, using the chain rule}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=\left (\frac {d}{d t}\frac {d}{d t}y \left (t \right )\right ) {t^{\prime }\left (x \right )}^{2}+\left (\frac {d}{d x}t^{\prime }\left (x \right )\right ) \left (\frac {d}{d t}y \left (t \right )\right ) \\ {} & \circ & \textrm {Compute derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=\frac {\frac {d}{d t}\frac {d}{d t}y \left (t \right )}{x^{2}}-\frac {\frac {d}{d t}y \left (t \right )}{x^{2}} \\ {} & \circ & \textrm {Calculate the}\hspace {3pt} \hspace {3pt}\textrm {3rd}\hspace {3pt} \hspace {3pt}\textrm {derivative of}\hspace {3pt} \hspace {3pt}\textrm {y}\hspace {3pt} \hspace {3pt}\textrm {with respect to}\hspace {3pt} \hspace {3pt}\textrm {x}\hspace {3pt} \hspace {3pt}\textrm {, using the chain rule}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }=\left (\frac {d}{d t}\frac {d^{2}}{d t^{2}}y \left (t \right )\right ) {t^{\prime }\left (x \right )}^{3}+3 t^{\prime }\left (x \right ) \left (\frac {d}{d x}t^{\prime }\left (x \right )\right ) \left (\frac {d}{d t}\frac {d}{d t}y \left (t \right )\right )+\left (\frac {d}{d x}t^{\prime \prime }\left (x \right )\right ) \left (\frac {d}{d t}y \left (t \right )\right ) \\ {} & \circ & \textrm {Compute derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime \prime }=\frac {\frac {d}{d t}\frac {d^{2}}{d t^{2}}y \left (t \right )}{x^{3}}-\frac {3 \left (\frac {d}{d t}\frac {d}{d t}y \left (t \right )\right )}{x^{3}}+\frac {2 \left (\frac {d}{d t}y \left (t \right )\right )}{x^{3}} \\ & {} & \textrm {Substitute the change of variables back into the ODE}\hspace {3pt} \\ {} & {} & \left (\frac {\frac {d}{d t}\frac {d^{2}}{d t^{2}}y \left (t \right )}{x^{3}}-\frac {3 \left (\frac {d}{d t}\frac {d}{d t}y \left (t \right )\right )}{x^{3}}+\frac {2 \left (\frac {d}{d t}y \left (t \right )\right )}{x^{3}}\right ) x^{3}+\left (\frac {\frac {d}{d t}\frac {d}{d t}y \left (t \right )}{x^{2}}-\frac {\frac {d}{d t}y \left (t \right )}{x^{2}}\right ) x^{2}+\frac {d}{d t}y \left (t \right )+y \left (t \right )=0 \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & \frac {d}{d t}\frac {d^{2}}{d t^{2}}y \left (t \right )-2 \frac {d}{d t}\frac {d}{d t}y \left (t \right )+2 \frac {d}{d t}y \left (t \right )+y \left (t \right )=0 \\ \square & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{1}\left (t \right ) \\ {} & {} & y_{1}\left (t \right )=y \left (t \right ) \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{2}\left (t \right ) \\ {} & {} & y_{2}\left (t \right )=\frac {d}{d t}y \left (t \right ) \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{3}\left (t \right ) \\ {} & {} & y_{3}\left (t \right )=\frac {d}{d t}\frac {d}{d t}y \left (t \right ) \\ {} & \circ & \textrm {Isolate for}\hspace {3pt} \frac {d}{d t}y_{3}\left (t \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & \frac {d}{d t}y_{3}\left (t \right )=2 y_{3}\left (t \right )-2 y_{2}\left (t \right )-y_{1}\left (t \right ) \\ & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & {} & \left [y_{2}\left (t \right )=\frac {d}{d t}y_{1}\left (t \right ), y_{3}\left (t \right )=\frac {d}{d t}y_{2}\left (t \right ), \frac {d}{d t}y_{3}\left (t \right )=2 y_{3}\left (t \right )-2 y_{2}\left (t \right )-y_{1}\left (t \right )\right ] \\ \bullet & {} & \textrm {Define vector}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{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 {System to solve}\hspace {3pt} \\ {} & {} & \frac {d}{d t}{\moverset {\rightarrow }{y}}\left (t \right )=\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -1 & -2 & 2 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (t \right ) \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -1 & -2 & 2 \end {array}\right ] \\ \bullet & {} & \textrm {Rewrite the system as}\hspace {3pt} \\ {} & {} & \frac {d}{d t}{\moverset {\rightarrow }{y}}\left (t \right )=A \cdot {\moverset {\rightarrow }{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 (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}, \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )^{2}} \\ \frac {1}{-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}} \\ 1 \end {array}\right ]\right ], \left [\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {1}{\left (\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ]\right ], \left [\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {1}{\left (\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}, \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )^{2}} \\ \frac {1}{-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}={\mathrm e}^{\left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}\right ) t}\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )^{2}} \\ \frac {1}{-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {1}{\left (\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}\cdot \left [\begin {array}{c} \frac {1}{\left (\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Use Euler identity to write solution in terms of}\hspace {3pt} \sin \hspace {3pt}\textrm {and}\hspace {3pt} \cos \\ {} & {} & {\mathrm e}^{\left (\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}\right ) t}\cdot \left (\cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) t}{2}\right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}\right ) t}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) t}{2}\right )}{\left (\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {\cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) t}{2}\right )}{\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )}{2}} \\ \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\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 }{y}}_{2}\left (t \right )={\mathrm e}^{\left (\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}\right ) t}\cdot \left [\begin {array}{c} -\frac {18 \left (188+12 \sqrt {249}\right )^{\frac {2}{3}} \left (-\sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \left (188+12 \sqrt {249}\right )^{\frac {4}{3}}+\cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \left (188+12 \sqrt {249}\right )^{\frac {4}{3}}-64 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}-96 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {249}+64 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}-96 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {249}-1440 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}-1440 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )\right )}{\left (\left (188+12 \sqrt {249}\right )^{\frac {4}{3}}+816+48 \sqrt {249}+24 \left (188+12 \sqrt {249}\right )^{\frac {2}{3}}-32 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}\right )^{2}} \\ \frac {3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}} \left (\sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+\cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}+8 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}-8 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )\right )}{\left (188+12 \sqrt {249}\right )^{\frac {4}{3}}+816+48 \sqrt {249}+24 \left (188+12 \sqrt {249}\right )^{\frac {2}{3}}-32 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}} \\ \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) t}{2}\right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{3}\left (t \right )={\mathrm e}^{\left (\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}\right ) t}\cdot \left [\begin {array}{c} -\frac {18 \left (188+12 \sqrt {249}\right )^{\frac {2}{3}} \left (\cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \left (188+12 \sqrt {249}\right )^{\frac {4}{3}}+\sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \left (188+12 \sqrt {249}\right )^{\frac {4}{3}}+64 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}+96 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {249}+1440 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+64 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}-96 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {249}-1440 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )\right )}{\left (\left (188+12 \sqrt {249}\right )^{\frac {4}{3}}+816+48 \sqrt {249}+24 \left (188+12 \sqrt {249}\right )^{\frac {2}{3}}-32 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}\right )^{2}} \\ -\frac {3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}} \left (\cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \left (188+12 \sqrt {249}\right )^{\frac {2}{3}}-\sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}-8 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}+8 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )\right )}{\left (188+12 \sqrt {249}\right )^{\frac {4}{3}}+816+48 \sqrt {249}+24 \left (188+12 \sqrt {249}\right )^{\frac {2}{3}}-32 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}} \\ -\sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\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 }{y}}=c_{1} {\moverset {\rightarrow }{y}}_{1}+c_{2} {\moverset {\rightarrow }{y}}_{2}\left (t \right )+c_{3} {\moverset {\rightarrow }{y}}_{3}\left (t \right ) \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}=c_{1} {\mathrm e}^{\left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}\right ) t}\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}\right )^{2}} \\ \frac {1}{-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}} \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{\left (\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}\right ) t}\cdot \left [\begin {array}{c} -\frac {18 \left (188+12 \sqrt {249}\right )^{\frac {2}{3}} \left (-\sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \left (188+12 \sqrt {249}\right )^{\frac {4}{3}}+\cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \left (188+12 \sqrt {249}\right )^{\frac {4}{3}}-64 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}-96 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {249}+64 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}-96 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {249}-1440 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}-1440 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )\right )}{\left (\left (188+12 \sqrt {249}\right )^{\frac {4}{3}}+816+48 \sqrt {249}+24 \left (188+12 \sqrt {249}\right )^{\frac {2}{3}}-32 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}\right )^{2}} \\ \frac {3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}} \left (\sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+\cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}+8 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}-8 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )\right )}{\left (188+12 \sqrt {249}\right )^{\frac {4}{3}}+816+48 \sqrt {249}+24 \left (188+12 \sqrt {249}\right )^{\frac {2}{3}}-32 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}} \\ \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) t}{2}\right ) \end {array}\right ]+c_{3} {\mathrm e}^{\left (\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}+\frac {2}{3}\right ) t}\cdot \left [\begin {array}{c} -\frac {18 \left (188+12 \sqrt {249}\right )^{\frac {2}{3}} \left (\cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \left (188+12 \sqrt {249}\right )^{\frac {4}{3}}+\sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \left (188+12 \sqrt {249}\right )^{\frac {4}{3}}+64 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}+96 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {249}+1440 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+64 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}-96 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {249}-1440 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )\right )}{\left (\left (188+12 \sqrt {249}\right )^{\frac {4}{3}}+816+48 \sqrt {249}+24 \left (188+12 \sqrt {249}\right )^{\frac {2}{3}}-32 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}\right )^{2}} \\ -\frac {3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}} \left (\cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \left (188+12 \sqrt {249}\right )^{\frac {2}{3}}-\sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \sqrt {3}-8 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}+8 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right )\right )}{\left (188+12 \sqrt {249}\right )^{\frac {4}{3}}+816+48 \sqrt {249}+24 \left (188+12 \sqrt {249}\right )^{\frac {2}{3}}-32 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}} \\ -\sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}\right ) t}{2}\right ) \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y \left (t \right )=\frac {96 \left (\left (\left (\left (\left (-\frac {7 c_{3}}{32}-\frac {7 \sqrt {3}\, c_{2}}{96}\right ) \sqrt {83}-\frac {115 c_{2}}{96}-\frac {115 \sqrt {3}\, c_{3}}{96}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}+c_{2} \left (\frac {47}{3}+\sqrt {3}\, \sqrt {83}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}+\left (10 c_{3} -\frac {10 \sqrt {3}\, c_{2}}{3}\right ) \sqrt {83}+\frac {158 \sqrt {3}\, c_{3}}{3}-\frac {158 c_{2}}{3}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}}\right )+\left (\left (\left (\frac {7 c_{2}}{32}-\frac {7 \sqrt {3}\, c_{3}}{96}\right ) \sqrt {83}-\frac {115 c_{3}}{96}+\frac {115 \sqrt {3}\, c_{2}}{96}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}+c_{3} \left (\frac {47}{3}+\sqrt {3}\, \sqrt {83}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}+\left (-10 c_{2} -\frac {10 \sqrt {3}\, c_{3}}{3}\right ) \sqrt {83}-\frac {158 \sqrt {3}\, c_{2}}{3}-\frac {158 c_{3}}{3}\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}+8\right ) t}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}}\right )\right ) {\mathrm e}^{\frac {\left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}-8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}}+c_{1} \left (\left (\frac {7 \sqrt {3}\, \sqrt {83}}{48}+\frac {115}{48}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}+\left (\frac {47}{3}+\sqrt {3}\, \sqrt {83}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}+\frac {20 \sqrt {3}\, \sqrt {83}}{3}+\frac {316}{3}\right ) {\mathrm e}^{-\frac {\left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}-4 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}-8\right ) t}{6 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}}{13536 \sqrt {3}\, \sqrt {83}+213600} \\ \bullet & {} & \textrm {Change variables back using}\hspace {3pt} t =\ln \left (x \right ) \\ {} & {} & y=\frac {96 \left (\left (\left (\left (\left (-\frac {7 c_{3}}{32}-\frac {7 \sqrt {3}\, c_{2}}{96}\right ) \sqrt {83}-\frac {115 c_{2}}{96}-\frac {115 \sqrt {3}\, c_{3}}{96}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}+c_{2} \left (\frac {47}{3}+\sqrt {3}\, \sqrt {83}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}+\left (10 c_{3} -\frac {10 \sqrt {3}\, c_{2}}{3}\right ) \sqrt {83}+\frac {158 \sqrt {3}\, c_{3}}{3}-\frac {158 c_{2}}{3}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}}\right )+\left (\left (\left (\frac {7 c_{2}}{32}-\frac {7 \sqrt {3}\, c_{3}}{96}\right ) \sqrt {83}-\frac {115 c_{3}}{96}+\frac {115 \sqrt {3}\, c_{2}}{96}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}+c_{3} \left (\frac {47}{3}+\sqrt {3}\, \sqrt {83}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}+\left (-10 c_{2} -\frac {10 \sqrt {3}\, c_{3}}{3}\right ) \sqrt {83}-\frac {158 \sqrt {3}\, c_{2}}{3}-\frac {158 c_{3}}{3}\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}}\right )\right ) {\mathrm e}^{\frac {\left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}-8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}}+c_{1} \left (\left (\frac {7 \sqrt {3}\, \sqrt {83}}{48}+\frac {115}{48}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}+\left (\frac {47}{3}+\sqrt {3}\, \sqrt {83}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}+\frac {20 \sqrt {3}\, \sqrt {83}}{3}+\frac {316}{3}\right ) {\mathrm e}^{-\frac {\left (\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}-4 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}-8\right ) \ln \left (x \right )}{6 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}}{13536 \sqrt {3}\, \sqrt {83}+213600} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y=-\frac {7 \left (x^{\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}+8 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}-8}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}}} \left (\left (\left (\sqrt {3}\, c_{2} +3 c_{3} \right ) \sqrt {83}+\frac {115 c_{2}}{7}+\frac {115 \sqrt {3}\, c_{3}}{7}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}-\frac {96 c_{2} \left (\frac {47}{3}+\sqrt {3}\, \sqrt {83}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}}{7}+\frac {320 \left (\sqrt {3}\, c_{2} -3 c_{3} \right ) \sqrt {83}}{7}-\frac {5056 \sqrt {3}\, c_{3}}{7}+\frac {5056 c_{2}}{7}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}}\right )-\frac {115 x^{\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}+8 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}-8}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}}} \left (\left (\frac {7 \left (-\sqrt {3}\, c_{3} +3 c_{2} \right ) \sqrt {83}}{115}+\sqrt {3}\, c_{2} -c_{3} \right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}+\frac {96 c_{3} \left (\frac {47}{3}+\sqrt {3}\, \sqrt {83}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}}{115}+\frac {64 \left (-\sqrt {3}\, c_{3} -3 c_{2} \right ) \sqrt {83}}{23}-\frac {5056 \sqrt {3}\, c_{2}}{115}-\frac {5056 c_{3}}{115}\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}}\right )}{7}-2 c_{1} x^{-\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}-4 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}-8}{6 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}}} \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}} \sqrt {3}\, \sqrt {83}+\frac {48 \sqrt {3}\, \sqrt {83}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}}{7}+\frac {320 \sqrt {3}\, \sqrt {83}}{7}+\frac {115 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}}{7}+\frac {752 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}}{7}+\frac {5056}{7}\right )\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}}{96 \left (141 \sqrt {3}\, \sqrt {83}+2225\right )} \end {array} \]

Maple trace

`Methods for third order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
checking if the LODE has constant coefficients 
checking if the LODE is of Euler type 
<- LODE of Euler type successful`

\[ y \left (x \right ) = c_{1} x^{-\frac {\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}-4 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}-8}{6 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}}+c_{2} x^{\frac {-8+\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}}\right )+c_{3} x^{\frac {-8+\left (188+12 \sqrt {249}\right )^{\frac {2}{3}}+8 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}{12 \left (188+12 \sqrt {249}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {2}{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{\frac {1}{3}}}\right ) \]

\[ y(x)\to c_1 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}+1\&,1\right ]}+c_3 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}+1\&,3\right ]}+c_2 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}+1\&,2\right ]} \]

3.18 problem 18

3.18.1 Maple step by step solution