problem 18

And since \(x^{\lambda }\neq 0\) then dividing through by \(x^{\lambda }\), the above becomes

\[ \lambda +\lambda \left (\lambda -1\right )+\lambda \left (\lambda -1\right ) \left (\lambda -2\right )+1 = 0 \]

\[ \lambda ^{3}-2 \lambda ^{2}+2 \lambda +1 = 0 \]

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

\[ y = c_1 \,x^{-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}}+x^{\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}} \left (c_2 \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \ln \left (x \right )}{2}\right )-c_3 \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \ln \left (x \right )}{2}\right )\right ) \]

3.18.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{4} \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 )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}, \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}\right )^{2}} \\ \frac {1}{-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}} \\ 1 \end {array}\right ]\right ], \left [\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )}{2}, \left [\begin {array}{c} \frac {1}{\left (\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )}{2}} \\ 1 \end {array}\right ]\right ], \left [\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )}{2}, \left [\begin {array}{c} \frac {1}{\left (\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )}{2}} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}, \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}\right )^{2}} \\ \frac {1}{-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{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 )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}\right ) t}\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}\right )^{2}} \\ \frac {1}{-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{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 )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )}{2}, \left [\begin {array}{c} \frac {1}{\left (\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{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 )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )}{2}\right ) t}\cdot \left [\begin {array}{c} \frac {1}{\left (\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{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 )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}\right ) t}\cdot \left (\cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) t}{2}\right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )}{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{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 )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) t}{2}\right )}{\left (\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )}{2}\right )^{2}} \\ \frac {\cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) t}{2}\right )}{\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )}{2}} \\ \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) t}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{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 )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}\right ) t}\cdot \left [\begin {array}{c} -\frac {18 \left (188+12 \sqrt {249}\right )^{{2}/{3}} \left (-\left (188+12 \sqrt {249}\right )^{{4}/{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}+\left (188+12 \sqrt {249}\right )^{{4}/{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )-64 \left (188+12 \sqrt {249}\right )^{{1}/{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}-96 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}\, \sqrt {249}+64 \left (188+12 \sqrt {249}\right )^{{1}/{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )-96 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {249}-1440 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}-1440 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )\right )}{\left (\left (188+12 \sqrt {249}\right )^{{4}/{3}}+816+48 \sqrt {249}+24 \left (188+12 \sqrt {249}\right )^{{2}/{3}}-32 \left (188+12 \sqrt {249}\right )^{{1}/{3}}\right )^{2}} \\ \frac {3 \left (188+12 \sqrt {249}\right )^{{1}/{3}} \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}+\left (188+12 \sqrt {249}\right )^{{2}/{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )+8 \left (188+12 \sqrt {249}\right )^{{1}/{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )+8 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}-8 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )\right )}{\left (188+12 \sqrt {249}\right )^{{4}/{3}}+816+48 \sqrt {249}+24 \left (188+12 \sqrt {249}\right )^{{2}/{3}}-32 \left (188+12 \sqrt {249}\right )^{{1}/{3}}} \\ \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{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 )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}\right ) t}\cdot \left [\begin {array}{c} -\frac {18 \left (188+12 \sqrt {249}\right )^{{2}/{3}} \left (\left (188+12 \sqrt {249}\right )^{{4}/{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}+\left (188+12 \sqrt {249}\right )^{{4}/{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )+64 \left (188+12 \sqrt {249}\right )^{{1}/{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}+96 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}\, \sqrt {249}+64 \left (188+12 \sqrt {249}\right )^{{1}/{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )-96 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {249}+1440 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}-1440 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )\right )}{\left (\left (188+12 \sqrt {249}\right )^{{4}/{3}}+816+48 \sqrt {249}+24 \left (188+12 \sqrt {249}\right )^{{2}/{3}}-32 \left (188+12 \sqrt {249}\right )^{{1}/{3}}\right )^{2}} \\ -\frac {3 \left (188+12 \sqrt {249}\right )^{{1}/{3}} \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}-\left (188+12 \sqrt {249}\right )^{{2}/{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )-8 \left (188+12 \sqrt {249}\right )^{{1}/{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )+8 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}+8 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )\right )}{\left (188+12 \sqrt {249}\right )^{{4}/{3}}+816+48 \sqrt {249}+24 \left (188+12 \sqrt {249}\right )^{{2}/{3}}-32 \left (188+12 \sqrt {249}\right )^{{1}/{3}}} \\ -\sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) t}{2}\right ) \end {array}\right ]\right ] \\ \bullet & {} & \textrm {General solution to the system of ODEs}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}=\mathit {C1} {\moverset {\rightarrow }{y}}_{1}+\mathit {C2} {\moverset {\rightarrow }{y}}_{2}\left (t \right )+\mathit {C3} {\moverset {\rightarrow }{y}}_{3}\left (t \right ) \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}=\mathit {C1} \,{\mathrm e}^{\left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}\right ) t}\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}\right )^{2}} \\ \frac {1}{-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}} \\ 1 \end {array}\right ]+\mathit {C2} \,{\mathrm e}^{\left (\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}\right ) t}\cdot \left [\begin {array}{c} -\frac {18 \left (188+12 \sqrt {249}\right )^{{2}/{3}} \left (-\left (188+12 \sqrt {249}\right )^{{4}/{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}+\left (188+12 \sqrt {249}\right )^{{4}/{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )-64 \left (188+12 \sqrt {249}\right )^{{1}/{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}-96 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}\, \sqrt {249}+64 \left (188+12 \sqrt {249}\right )^{{1}/{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )-96 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {249}-1440 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}-1440 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )\right )}{\left (\left (188+12 \sqrt {249}\right )^{{4}/{3}}+816+48 \sqrt {249}+24 \left (188+12 \sqrt {249}\right )^{{2}/{3}}-32 \left (188+12 \sqrt {249}\right )^{{1}/{3}}\right )^{2}} \\ \frac {3 \left (188+12 \sqrt {249}\right )^{{1}/{3}} \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}+\left (188+12 \sqrt {249}\right )^{{2}/{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )+8 \left (188+12 \sqrt {249}\right )^{{1}/{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )+8 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}-8 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )\right )}{\left (188+12 \sqrt {249}\right )^{{4}/{3}}+816+48 \sqrt {249}+24 \left (188+12 \sqrt {249}\right )^{{2}/{3}}-32 \left (188+12 \sqrt {249}\right )^{{1}/{3}}} \\ \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) t}{2}\right ) \end {array}\right ]+\mathit {C3} \,{\mathrm e}^{\left (\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}\right ) t}\cdot \left [\begin {array}{c} -\frac {18 \left (188+12 \sqrt {249}\right )^{{2}/{3}} \left (\left (188+12 \sqrt {249}\right )^{{4}/{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}+\left (188+12 \sqrt {249}\right )^{{4}/{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )+64 \left (188+12 \sqrt {249}\right )^{{1}/{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}+96 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}\, \sqrt {249}+64 \left (188+12 \sqrt {249}\right )^{{1}/{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )-96 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {249}+1440 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}-1440 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )\right )}{\left (\left (188+12 \sqrt {249}\right )^{{4}/{3}}+816+48 \sqrt {249}+24 \left (188+12 \sqrt {249}\right )^{{2}/{3}}-32 \left (188+12 \sqrt {249}\right )^{{1}/{3}}\right )^{2}} \\ -\frac {3 \left (188+12 \sqrt {249}\right )^{{1}/{3}} \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}-\left (188+12 \sqrt {249}\right )^{{2}/{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )-8 \left (188+12 \sqrt {249}\right )^{{1}/{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )+8 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \sqrt {3}+8 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )\right )}{\left (188+12 \sqrt {249}\right )^{{4}/{3}}+816+48 \sqrt {249}+24 \left (188+12 \sqrt {249}\right )^{{2}/{3}}-32 \left (188+12 \sqrt {249}\right )^{{1}/{3}}} \\ -\sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{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 \mathit {C3}}{32}-\frac {7 \sqrt {3}\, \mathit {C2}}{96}\right ) \sqrt {83}-\frac {115 \sqrt {3}\, \mathit {C3}}{96}-\frac {115 \mathit {C2}}{96}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+\mathit {C2} \left (\frac {47}{3}+\sqrt {3}\, \sqrt {83}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\left (-\frac {10 \sqrt {3}\, \mathit {C2}}{3}+10 \mathit {C3} \right ) \sqrt {83}+\frac {158 \sqrt {3}\, \mathit {C3}}{3}-\frac {158 \mathit {C2}}{3}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right )+\left (\left (\left (\frac {7 \mathit {C2}}{32}-\frac {7 \sqrt {3}\, \mathit {C3}}{96}\right ) \sqrt {83}+\frac {115 \sqrt {3}\, \mathit {C2}}{96}-\frac {115 \mathit {C3}}{96}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+\mathit {C3} \left (\frac {47}{3}+\sqrt {3}\, \sqrt {83}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\left (-\frac {10 \sqrt {3}\, \mathit {C3}}{3}-10 \mathit {C2} \right ) \sqrt {83}-\frac {158 \sqrt {3}\, \mathit {C2}}{3}-\frac {158 \mathit {C3}}{3}\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right )\right ) {\mathrm e}^{\frac {\left (-8+\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {249}\right )^{{1}/{3}}\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}}+\mathit {C1} \,{\mathrm e}^{-\frac {\left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}-4 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-8\right ) t}{6 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}} \left (\left (\frac {7 \sqrt {3}\, \sqrt {83}}{48}+\frac {115}{48}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+\left (\frac {47}{3}+\sqrt {3}\, \sqrt {83}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\frac {20 \sqrt {3}\, \sqrt {83}}{3}+\frac {316}{3}\right )\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{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 \mathit {C3}}{32}-\frac {7 \sqrt {3}\, \mathit {C2}}{96}\right ) \sqrt {83}-\frac {115 \sqrt {3}\, \mathit {C3}}{96}-\frac {115 \mathit {C2}}{96}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+\mathit {C2} \left (\frac {47}{3}+\sqrt {3}\, \sqrt {83}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\left (-\frac {10 \sqrt {3}\, \mathit {C2}}{3}+10 \mathit {C3} \right ) \sqrt {83}+\frac {158 \sqrt {3}\, \mathit {C3}}{3}-\frac {158 \mathit {C2}}{3}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right )+\left (\left (\left (\frac {7 \mathit {C2}}{32}-\frac {7 \sqrt {3}\, \mathit {C3}}{96}\right ) \sqrt {83}+\frac {115 \sqrt {3}\, \mathit {C2}}{96}-\frac {115 \mathit {C3}}{96}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+\mathit {C3} \left (\frac {47}{3}+\sqrt {3}\, \sqrt {83}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\left (-\frac {10 \sqrt {3}\, \mathit {C3}}{3}-10 \mathit {C2} \right ) \sqrt {83}-\frac {158 \sqrt {3}\, \mathit {C2}}{3}-\frac {158 \mathit {C3}}{3}\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right )\right ) {\mathrm e}^{\frac {\left (-8+\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {249}\right )^{{1}/{3}}\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}}+\mathit {C1} \,{\mathrm e}^{-\frac {\left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}-4 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-8\right ) \ln \left (x \right )}{6 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}} \left (\left (\frac {7 \sqrt {3}\, \sqrt {83}}{48}+\frac {115}{48}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+\left (\frac {47}{3}+\sqrt {3}\, \sqrt {83}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\frac {20 \sqrt {3}\, \sqrt {83}}{3}+\frac {316}{3}\right )\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{13536 \sqrt {3}\, \sqrt {83}+213600} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y=-\frac {7 \left (\left (\left (\left (\sqrt {3}\, \mathit {C2} +3 \mathit {C3} \right ) \sqrt {83}+\frac {115 \mathit {C2}}{7}+\frac {115 \sqrt {3}\, \mathit {C3}}{7}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}-\frac {96 \mathit {C2} \left (\frac {47}{3}+\sqrt {3}\, \sqrt {83}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}{7}+\frac {320 \left (\sqrt {3}\, \mathit {C2} -3 \mathit {C3} \right ) \sqrt {83}}{7}+\frac {5056 \mathit {C2}}{7}-\frac {5056 \sqrt {3}\, \mathit {C3}}{7}\right ) x^{\frac {-8+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right )-\frac {115 \left (\left (\frac {7 \left (-\sqrt {3}\, \mathit {C3} +3 \mathit {C2} \right ) \sqrt {83}}{115}+\sqrt {3}\, \mathit {C2} -\mathit {C3} \right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+\frac {96 \mathit {C3} \left (\frac {47}{3}+\sqrt {3}\, \sqrt {83}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}{115}+\frac {64 \left (-\sqrt {3}\, \mathit {C3} -3 \mathit {C2} \right ) \sqrt {83}}{23}-\frac {5056 \sqrt {3}\, \mathit {C2}}{115}-\frac {5056 \mathit {C3}}{115}\right ) x^{\frac {-8+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right )}{7}-2 \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}} \sqrt {3}\, \sqrt {83}+\frac {48 \sqrt {83}\, \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}{7}+\frac {320 \sqrt {3}\, \sqrt {83}}{7}+\frac {115 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{7}+\frac {752 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}{7}+\frac {5056}{7}\right ) \mathit {C1} \,x^{-\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}-4 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-8}{6 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{96 \left (141 \sqrt {3}\, \sqrt {83}+2225\right )} \end {array} \]


root	multiplicity	type of root

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

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

3.18 problem 18

3.18.1 Solved as higher order Euler type ode

3.18.2 Maple step by step solution

3.18.3 Maple trace

3.18.4 Maple dsolve solution

3.18.5 Mathematica DSolve solution