2.3.18 problem 18
Internal
problem
ID
[8552]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
3.0
Problem
number
:
18
Date
solved
:
Thursday, December 12, 2024 at 09:30:02 AM
CAS
classification
:
[[_3rd_order, _with_linear_symmetries]]
Solve
\begin{align*} x^{4} y^{\prime \prime \prime }+x^{3} y^{\prime \prime }+x^{2} y^{\prime }+x y&=0 \end{align*}
Solved as higher order Euler type ode
Time used: 0.169 (sec)
The ode can be normalized and rewritten as Euler ode.
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
\[ x^{4} y^{\prime \prime \prime }+x^{3} y^{\prime \prime }+x^{2} y^{\prime }+x y = 0 \]
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 )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}\\ \lambda _2 &= \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 {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}\\ \lambda _3 &= \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 {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} \end{align*}
This table summarises the result
| | |
| root |
multiplicity |
type of 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 |
| | |
| \(-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}\) | \(1\) | real root |
| | |
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
\[ y = x^{\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}} \left (c_1 \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \ln \left (x \right )}{2}\right )-c_2 \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \ln \left (x \right )}{2}\right )\right )+c_3 \,x^{-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}} \]
The fundamental set of solutions for the homogeneous solution are the following
\begin{align*}
y_1 &= 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}} \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 ) \\
y_2 &= -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}} \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 ) \\
y_3 &= 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}} \\
\end{align*}
Maple step by step solution
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{4} \left (\frac {d^{3}}{d x^{3}}y \left (x \right )\right )+\left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right ) x^{3}+x^{2} \left (\frac {d}{d x}y \left (x \right )\right )+x y \left (x \right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d^{3}}{d x^{3}}y \left (x \right ) \\ \bullet & {} & \textrm {Isolate 3rd derivative}\hspace {3pt} \\ {} & {} & \frac {d^{3}}{d x^{3}}y \left (x \right )=-\frac {y \left (x \right )}{x^{3}}-\frac {\left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right ) x +\frac {d}{d x}y \left (x \right )}{x^{2}} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y \left (x \right )\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^{3}}{d x^{3}}y \left (x \right )+\frac {\frac {d^{2}}{d x^{2}}y \left (x \right )}{x}+\frac {\frac {d}{d x}y \left (x \right )}{x^{2}}+\frac {y \left (x \right )}{x^{3}}=0 \\ \bullet & {} & \textrm {Multiply by denominators of the ODE}\hspace {3pt} \\ {} & {} & x^{3} \left (\frac {d^{3}}{d x^{3}}y \left (x \right )\right )+x^{2} \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )+x \left (\frac {d}{d x}y \left (x \right )\right )+y \left (x \right )=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} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\left (\frac {d}{d t}y \left (t \right )\right ) \left (\frac {d}{d x}t \left (x \right )\right ) \\ {} & \circ & \textrm {Compute derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\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^{2}}{d x^{2}}y \left (x \right )=\left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right ) \left (\frac {d}{d x}t \left (x \right )\right )^{2}+\left (\frac {d^{2}}{d x^{2}}t \left (x \right )\right ) \left (\frac {d}{d t}y \left (t \right )\right ) \\ {} & \circ & \textrm {Compute derivative}\hspace {3pt} \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right )=\frac {\frac {d^{2}}{d t^{2}}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^{3}}{d x^{3}}y \left (x \right )=\left (\frac {d^{3}}{d t^{3}}y \left (t \right )\right ) \left (\frac {d}{d x}t \left (x \right )\right )^{3}+3 \left (\frac {d}{d x}t \left (x \right )\right ) \left (\frac {d^{2}}{d x^{2}}t \left (x \right )\right ) \left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right )+\left (\frac {d^{3}}{d x^{3}}t \left (x \right )\right ) \left (\frac {d}{d t}y \left (t \right )\right ) \\ {} & \circ & \textrm {Compute derivative}\hspace {3pt} \\ {} & {} & \frac {d^{3}}{d x^{3}}y \left (x \right )=\frac {\frac {d^{3}}{d t^{3}}y \left (t \right )}{x^{3}}-\frac {3 \left (\frac {d^{2}}{d t^{2}}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} \\ {} & {} & x^{3} \left (\frac {\frac {d^{3}}{d t^{3}}y \left (t \right )}{x^{3}}-\frac {3 \left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right )}{x^{3}}+\frac {2 \left (\frac {d}{d t}y \left (t \right )\right )}{x^{3}}\right )+x^{2} \left (\frac {\frac {d^{2}}{d t^{2}}y \left (t \right )}{x^{2}}-\frac {\frac {d}{d t}y \left (t \right )}{x^{2}}\right )+\frac {d}{d t}y \left (t \right )+y \left (t \right )=0 \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & \frac {d^{3}}{d t^{3}}y \left (t \right )-2 \frac {d^{2}}{d t^{2}}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^{2}}{d t^{2}}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 (\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 )-\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 \sqrt {3}\, \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 {3}\, \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}-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 \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 (\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 )+\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 \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 \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 \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 (\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 )+\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 \sqrt {3}\, \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 {3}\, \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}+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 \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 (\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 )-\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 \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 \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 \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 (\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 )-\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 \sqrt {3}\, \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 {3}\, \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}-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 \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 (\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 )+\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 \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 \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 \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 (\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 )+\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 \sqrt {3}\, \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 {3}\, \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}+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 \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 (\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 )-\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 \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 \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 \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 (188+12 \sqrt {83}\, \sqrt {3}\right )^{{2}/{3}} \left (\left (\left (\left (\left (-\frac {7 \sqrt {3}\, \mathit {C2}}{96}-\frac {7 \mathit {C3}}{32}\right ) \sqrt {83}-\frac {115 \mathit {C2}}{96}-\frac {115 \mathit {C3} \sqrt {3}}{96}\right ) \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{2}/{3}}+\mathit {C2} \left (\sqrt {83}\, \sqrt {3}+\frac {47}{3}\right ) \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{1}/{3}}+\left (10 \mathit {C3} -\frac {10 \sqrt {3}\, \mathit {C2}}{3}\right ) \sqrt {83}+\frac {158 \mathit {C3} \sqrt {3}}{3}-\frac {158 \mathit {C2}}{3}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{1}/{3}}}\right )+\sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{1}/{3}}}\right ) \left (\left (\left (-\frac {7 \mathit {C3} \sqrt {3}}{96}+\frac {7 \mathit {C2}}{32}\right ) \sqrt {83}-\frac {115 \mathit {C3}}{96}+\frac {115 \sqrt {3}\, \mathit {C2}}{96}\right ) \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{2}/{3}}+\mathit {C3} \left (\sqrt {83}\, \sqrt {3}+\frac {47}{3}\right ) \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{1}/{3}}+\left (-10 \mathit {C2} -\frac {10 \mathit {C3} \sqrt {3}}{3}\right ) \sqrt {83}-\frac {158 \sqrt {3}\, \mathit {C2}}{3}-\frac {158 \mathit {C3}}{3}\right )\right ) {\mathrm e}^{\frac {\left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}}+\left (\left (\frac {7 \sqrt {83}\, \sqrt {3}}{48}+\frac {115}{48}\right ) \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{2}/{3}}+\left (\sqrt {83}\, \sqrt {3}+\frac {47}{3}\right ) \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{1}/{3}}+\frac {316}{3}+\frac {20 \sqrt {83}\, \sqrt {3}}{3}\right ) {\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}}}} \mathit {C1} \right )}{13536 \sqrt {83}\, \sqrt {3}+213600} \\ \bullet & {} & \textrm {Change variables back using}\hspace {3pt} t =\ln \left (x \right ) \\ {} & {} & y \left (x \right )=\frac {96 \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{2}/{3}} \left (\left (\left (\left (\left (-\frac {7 \sqrt {3}\, \mathit {C2}}{96}-\frac {7 \mathit {C3}}{32}\right ) \sqrt {83}-\frac {115 \mathit {C2}}{96}-\frac {115 \mathit {C3} \sqrt {3}}{96}\right ) \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{2}/{3}}+\mathit {C2} \left (\sqrt {83}\, \sqrt {3}+\frac {47}{3}\right ) \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{1}/{3}}+\left (10 \mathit {C3} -\frac {10 \sqrt {3}\, \mathit {C2}}{3}\right ) \sqrt {83}+\frac {158 \mathit {C3} \sqrt {3}}{3}-\frac {158 \mathit {C2}}{3}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{1}/{3}}}\right )+\sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{1}/{3}}}\right ) \left (\left (\left (-\frac {7 \mathit {C3} \sqrt {3}}{96}+\frac {7 \mathit {C2}}{32}\right ) \sqrt {83}-\frac {115 \mathit {C3}}{96}+\frac {115 \sqrt {3}\, \mathit {C2}}{96}\right ) \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{2}/{3}}+\mathit {C3} \left (\sqrt {83}\, \sqrt {3}+\frac {47}{3}\right ) \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{1}/{3}}+\left (-10 \mathit {C2} -\frac {10 \mathit {C3} \sqrt {3}}{3}\right ) \sqrt {83}-\frac {158 \sqrt {3}\, \mathit {C2}}{3}-\frac {158 \mathit {C3}}{3}\right )\right ) {\mathrm e}^{\frac {\left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}}+\left (\left (\frac {7 \sqrt {83}\, \sqrt {3}}{48}+\frac {115}{48}\right ) \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{2}/{3}}+\left (\sqrt {83}\, \sqrt {3}+\frac {47}{3}\right ) \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{1}/{3}}+\frac {316}{3}+\frac {20 \sqrt {83}\, \sqrt {3}}{3}\right ) {\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}}}} \mathit {C1} \right )}{13536 \sqrt {83}\, \sqrt {3}+213600} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y \left (x \right )=-\frac {7 \left (\left (\left (\left (\sqrt {3}\, \mathit {C2} +3 \mathit {C3} \right ) \sqrt {83}+\frac {115 \mathit {C2}}{7}+\frac {115 \mathit {C3} \sqrt {3}}{7}\right ) \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{2}/{3}}-\frac {96 \mathit {C2} \left (\sqrt {83}\, \sqrt {3}+\frac {47}{3}\right ) \left (188+12 \sqrt {83}\, \sqrt {3}\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 \mathit {C3} \sqrt {3}}{7}\right ) x^{\frac {\left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{1}/{3}}-8}{12 \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{1}/{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{1}/{3}}}\right )-\frac {115 \left (\left (\frac {7 \left (-\mathit {C3} \sqrt {3}+3 \mathit {C2} \right ) \sqrt {83}}{115}+\sqrt {3}\, \mathit {C2} -\mathit {C3} \right ) \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{2}/{3}}+\frac {96 \mathit {C3} \left (\sqrt {83}\, \sqrt {3}+\frac {47}{3}\right ) \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{1}/{3}}}{115}+\frac {64 \left (-\mathit {C3} \sqrt {3}-3 \mathit {C2} \right ) \sqrt {83}}{23}-\frac {5056 \sqrt {3}\, \mathit {C2}}{115}-\frac {5056 \mathit {C3}}{115}\right ) x^{\frac {\left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{1}/{3}}-8}{12 \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{1}/{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{1}/{3}}}\right )}{7}-2 x^{-\frac {\left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{2}/{3}}-4 \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{1}/{3}}-8}{6 \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{1}/{3}}}} \left (\sqrt {83}\, \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{2}/{3}} \sqrt {3}+\frac {48 \sqrt {3}\, \sqrt {83}\, \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{1}/{3}}}{7}+\frac {320 \sqrt {83}\, \sqrt {3}}{7}+\frac {115 \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{2}/{3}}}{7}+\frac {752 \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{1}/{3}}}{7}+\frac {5056}{7}\right ) \mathit {C1} \right ) \left (188+12 \sqrt {83}\, \sqrt {3}\right )^{{2}/{3}}}{96 \left (141 \sqrt {83}\, \sqrt {3}+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`
Maple dsolve solution
Solving time : 0.006 (sec)
Leaf size : 184
dsolve(x^4*diff(diff(diff(y(x),x),x),x)+x^3*diff(diff(y(x),x),x)+diff(y(x),x)*x^2+x*y(x) = 0,
y(x),singsol=all)
\[
y = c_{1} x^{-\frac {\left (188+12 \sqrt {249}\right )^{{2}/{3}}-4 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-8}{6 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}}+c_{2} x^{\frac {-8+\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right )+c_3 \,x^{\frac {-8+\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right )
\]
Mathematica DSolve solution
Solving time : 0.005 (sec)
Leaf size : 81
DSolve[{x^4*D[y[x],{x,3}]+x^3*D[y[x],{x,2}]+x^2*D[y[x],x]+x*y[x]== 0,{}},
y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to c_1 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}+1\&,1\right ]}+c_3 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}+1\&,3\right ]}+c_2 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}+1\&,2\right ]}
\]