2.4 problem 11

Internal problem ID [825]
Internal file name [OUTPUT/825_Sunday_June_05_2022_01_50_37_AM_32825979/index.tex]

Book: Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section: Chapter 4.2, Higher order linear differential equations. Constant coefficients. page 180
Problem number: 11.
ODE order: 6.
ODE degree: 1.

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

Maple gives the following as the ode type

[[_high_order, _missing_x]]

\[ \boxed {y^{\left (6\right )}+y=0} \] The characteristic equation is \[ \lambda ^{6}+1 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= i\\ \lambda _2 &= -i\\ \lambda _3 &= \frac {\sqrt {-2 i \sqrt {3}+2}}{2}\\ \lambda _4 &= -\frac {\sqrt {-2 i \sqrt {3}+2}}{2}\\ \lambda _5 &= \frac {\sqrt {2+2 i \sqrt {3}}}{2}\\ \lambda _6 &= -\frac {\sqrt {2+2 i \sqrt {3}}}{2} \end {align*}

Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{-i x} c_{1} +{\mathrm e}^{i x} c_{2} +{\mathrm e}^{-\frac {\sqrt {-2 i \sqrt {3}+2}\, x}{2}} c_{3} +{\mathrm e}^{\frac {\sqrt {2+2 i \sqrt {3}}\, x}{2}} c_{4} +{\mathrm e}^{\frac {\sqrt {-2 i \sqrt {3}+2}\, x}{2}} c_{5} +{\mathrm e}^{-\frac {\sqrt {2+2 i \sqrt {3}}\, x}{2}} c_{6} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{-i x}\\ y_2 &= {\mathrm e}^{i x}\\ y_3 &= {\mathrm e}^{-\frac {\sqrt {-2 i \sqrt {3}+2}\, x}{2}}\\ y_4 &= {\mathrm e}^{\frac {\sqrt {2+2 i \sqrt {3}}\, x}{2}}\\ y_5 &= {\mathrm e}^{\frac {\sqrt {-2 i \sqrt {3}+2}\, x}{2}}\\ y_6 &= {\mathrm e}^{-\frac {\sqrt {2+2 i \sqrt {3}}\, x}{2}} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-i x} c_{1} +{\mathrm e}^{i x} c_{2} +{\mathrm e}^{-\frac {\sqrt {-2 i \sqrt {3}+2}\, x}{2}} c_{3} +{\mathrm e}^{\frac {\sqrt {2+2 i \sqrt {3}}\, x}{2}} c_{4} +{\mathrm e}^{\frac {\sqrt {-2 i \sqrt {3}+2}\, x}{2}} c_{5} +{\mathrm e}^{-\frac {\sqrt {2+2 i \sqrt {3}}\, x}{2}} c_{6} \\ \end{align*}

Verification of solutions

\[ y = {\mathrm e}^{-i x} c_{1} +{\mathrm e}^{i x} c_{2} +{\mathrm e}^{-\frac {\sqrt {-2 i \sqrt {3}+2}\, x}{2}} c_{3} +{\mathrm e}^{\frac {\sqrt {2+2 i \sqrt {3}}\, x}{2}} c_{4} +{\mathrm e}^{\frac {\sqrt {-2 i \sqrt {3}+2}\, x}{2}} c_{5} +{\mathrm e}^{-\frac {\sqrt {2+2 i \sqrt {3}}\, x}{2}} c_{6} \] Verified OK.

2.4.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\left (6\right )}+y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 6 \\ {} & {} & y^{\left (6\right )} \\ \square & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{1}\left (x \right ) \\ {} & {} & y_{1}\left (x \right )=y \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{2}\left (x \right ) \\ {} & {} & y_{2}\left (x \right )=y^{\prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{3}\left (x \right ) \\ {} & {} & y_{3}\left (x \right )=y^{\prime \prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{4}\left (x \right ) \\ {} & {} & y_{4}\left (x \right )=y^{\prime \prime \prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{5}\left (x \right ) \\ {} & {} & y_{5}\left (x \right )=y^{\prime \prime \prime \prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{6}\left (x \right ) \\ {} & {} & y_{6}\left (x \right )=y^{\left (5\right )} \\ {} & \circ & \textrm {Isolate for}\hspace {3pt} y_{6}^{\prime }\left (x \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & y_{6}^{\prime }\left (x \right )=-y_{1}\left (x \right ) \\ & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & {} & \left [y_{2}\left (x \right )=y_{1}^{\prime }\left (x \right ), y_{3}\left (x \right )=y_{2}^{\prime }\left (x \right ), y_{4}\left (x \right )=y_{3}^{\prime }\left (x \right ), y_{5}\left (x \right )=y_{4}^{\prime }\left (x \right ), y_{6}\left (x \right )=y_{5}^{\prime }\left (x \right ), y_{6}^{\prime }\left (x \right )=-y_{1}\left (x \right )\right ] \\ \bullet & {} & \textrm {Define vector}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}\left (x \right )=\left [\begin {array}{c} y_{1}\left (x \right ) \\ y_{2}\left (x \right ) \\ y_{3}\left (x \right ) \\ y_{4}\left (x \right ) \\ y_{5}\left (x \right ) \\ y_{6}\left (x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (x \right )=\left [\begin {array}{cccccc} 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 & 0 & 0 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (x \right ) \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{cccccc} 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 & 0 & 0 \end {array}\right ] \\ \bullet & {} & \textrm {Rewrite the system as}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (x \right )=A \cdot {\moverset {\rightarrow }{y}}\left (x \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 [\mathrm {-I}, \left [\begin {array}{c} \mathrm {I} \\ 1 \\ \mathrm {-I} \\ -1 \\ \mathrm {I} \\ 1 \end {array}\right ]\right ], \left [\mathrm {I}, \left [\begin {array}{c} \mathrm {-I} \\ 1 \\ \mathrm {I} \\ -1 \\ \mathrm {-I} \\ 1 \end {array}\right ]\right ], \left [-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{5}} \\ \frac {1}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{4}} \\ \frac {1}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{3}} \\ \frac {1}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{2}} \\ \frac {1}{-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}} \\ 1 \end {array}\right ]\right ], \left [\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}, \left [\begin {array}{c} \frac {1}{\left (\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{5}} \\ \frac {1}{\left (\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{4}} \\ \frac {1}{\left (\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{3}} \\ \frac {1}{\left (\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{2}} \\ \frac {1}{\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}} \\ 1 \end {array}\right ]\right ], \left [\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}, \left [\begin {array}{c} \frac {1}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{5}} \\ \frac {1}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{4}} \\ \frac {1}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{3}} \\ \frac {1}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{2}} \\ \frac {1}{\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}} \\ 1 \end {array}\right ]\right ], \left [\frac {\sqrt {3}}{2}+\frac {\mathrm {I}}{2}, \left [\begin {array}{c} \frac {1}{\left (\frac {\sqrt {3}}{2}+\frac {\mathrm {I}}{2}\right )^{5}} \\ \frac {1}{\left (\frac {\sqrt {3}}{2}+\frac {\mathrm {I}}{2}\right )^{4}} \\ \frac {1}{\left (\frac {\sqrt {3}}{2}+\frac {\mathrm {I}}{2}\right )^{3}} \\ \frac {1}{\left (\frac {\sqrt {3}}{2}+\frac {\mathrm {I}}{2}\right )^{2}} \\ \frac {1}{\frac {\sqrt {3}}{2}+\frac {\mathrm {I}}{2}} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [\mathrm {-I}, \left [\begin {array}{c} \mathrm {I} \\ 1 \\ \mathrm {-I} \\ -1 \\ \mathrm {I} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\mathrm {-I} x}\cdot \left [\begin {array}{c} \mathrm {I} \\ 1 \\ \mathrm {-I} \\ -1 \\ \mathrm {I} \\ 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 \\ {} & {} & \left (\cos \left (x \right )-\mathrm {I} \sin \left (x \right )\right )\cdot \left [\begin {array}{c} \mathrm {I} \\ 1 \\ \mathrm {-I} \\ -1 \\ \mathrm {I} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & \left [\begin {array}{c} \mathrm {I} \left (\cos \left (x \right )-\mathrm {I} \sin \left (x \right )\right ) \\ \cos \left (x \right )-\mathrm {I} \sin \left (x \right ) \\ \mathrm {-I} \left (\cos \left (x \right )-\mathrm {I} \sin \left (x \right )\right ) \\ -\cos \left (x \right )+\mathrm {I} \sin \left (x \right ) \\ \mathrm {I} \left (\cos \left (x \right )-\mathrm {I} \sin \left (x \right )\right ) \\ \cos \left (x \right )-\mathrm {I} \sin \left (x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{y}}_{1}\left (x \right )=\left [\begin {array}{c} \sin \left (x \right ) \\ \cos \left (x \right ) \\ -\sin \left (x \right ) \\ -\cos \left (x \right ) \\ \sin \left (x \right ) \\ \cos \left (x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{2}\left (x \right )=\left [\begin {array}{c} \cos \left (x \right ) \\ -\sin \left (x \right ) \\ -\cos \left (x \right ) \\ \sin \left (x \right ) \\ \cos \left (x \right ) \\ -\sin \left (x \right ) \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{5}} \\ \frac {1}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{4}} \\ \frac {1}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{3}} \\ \frac {1}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{2}} \\ \frac {1}{-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{5}} \\ \frac {1}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{4}} \\ \frac {1}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{3}} \\ \frac {1}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{2}} \\ \frac {1}{-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{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}^{-\frac {\sqrt {3}\, x}{2}}\cdot \left (\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{5}} \\ \frac {1}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{4}} \\ \frac {1}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{3}} \\ \frac {1}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{2}} \\ \frac {1}{-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{-\frac {\sqrt {3}\, x}{2}}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{5}} \\ \frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{4}} \\ \frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{3}} \\ \frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{2}} \\ \frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}} \\ \cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{y}}_{3}\left (x \right )={\mathrm e}^{-\frac {\sqrt {3}\, x}{2}}\cdot \left [\begin {array}{c} \Re \left (\frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{5}}\right ) \\ \Re \left (\frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{4}}\right ) \\ \sin \left (\frac {x}{2}\right ) \\ \frac {\cos \left (\frac {x}{2}\right )}{2}-\frac {\sqrt {3}\, \sin \left (\frac {x}{2}\right )}{2} \\ -\frac {\cos \left (\frac {x}{2}\right ) \sqrt {3}}{2}+\frac {\sin \left (\frac {x}{2}\right )}{2} \\ \cos \left (\frac {x}{2}\right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{4}\left (x \right )={\mathrm e}^{-\frac {\sqrt {3}\, x}{2}}\cdot \left [\begin {array}{c} \Im \left (\frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{5}}\right ) \\ \Im \left (\frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{4}}\right ) \\ \cos \left (\frac {x}{2}\right ) \\ -\frac {\cos \left (\frac {x}{2}\right ) \sqrt {3}}{2}-\frac {\sin \left (\frac {x}{2}\right )}{2} \\ \frac {\cos \left (\frac {x}{2}\right )}{2}+\frac {\sqrt {3}\, \sin \left (\frac {x}{2}\right )}{2} \\ -\sin \left (\frac {x}{2}\right ) \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}, \left [\begin {array}{c} \frac {1}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{5}} \\ \frac {1}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{4}} \\ \frac {1}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{3}} \\ \frac {1}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{2}} \\ \frac {1}{\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{5}} \\ \frac {1}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{4}} \\ \frac {1}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{3}} \\ \frac {1}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{2}} \\ \frac {1}{\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{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}^{\frac {\sqrt {3}\, x}{2}}\cdot \left (\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{5}} \\ \frac {1}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{4}} \\ \frac {1}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{3}} \\ \frac {1}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{2}} \\ \frac {1}{\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\frac {\sqrt {3}\, x}{2}}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{5}} \\ \frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{4}} \\ \frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{3}} \\ \frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{2}} \\ \frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}} \\ \cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{y}}_{5}\left (x \right )={\mathrm e}^{\frac {\sqrt {3}\, x}{2}}\cdot \left [\begin {array}{c} \Re \left (\frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{5}}\right ) \\ \Re \left (\frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{4}}\right ) \\ \sin \left (\frac {x}{2}\right ) \\ \frac {\cos \left (\frac {x}{2}\right )}{2}+\frac {\sqrt {3}\, \sin \left (\frac {x}{2}\right )}{2} \\ \frac {\cos \left (\frac {x}{2}\right ) \sqrt {3}}{2}+\frac {\sin \left (\frac {x}{2}\right )}{2} \\ \cos \left (\frac {x}{2}\right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{6}\left (x \right )={\mathrm e}^{\frac {\sqrt {3}\, x}{2}}\cdot \left [\begin {array}{c} \Im \left (\frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{5}}\right ) \\ \Im \left (\frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{4}}\right ) \\ \cos \left (\frac {x}{2}\right ) \\ \frac {\cos \left (\frac {x}{2}\right ) \sqrt {3}}{2}-\frac {\sin \left (\frac {x}{2}\right )}{2} \\ \frac {\cos \left (\frac {x}{2}\right )}{2}-\frac {\sqrt {3}\, \sin \left (\frac {x}{2}\right )}{2} \\ -\sin \left (\frac {x}{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}\left (x \right )+c_{2} {\moverset {\rightarrow }{y}}_{2}\left (x \right )+c_{3} {\moverset {\rightarrow }{y}}_{3}\left (x \right )+c_{4} {\moverset {\rightarrow }{y}}_{4}\left (x \right )+c_{5} {\moverset {\rightarrow }{y}}_{5}\left (x \right )+c_{6} {\moverset {\rightarrow }{y}}_{6}\left (x \right ) \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}=c_{3} {\mathrm e}^{-\frac {\sqrt {3}\, x}{2}}\cdot \left [\begin {array}{c} \Re \left (\frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{5}}\right ) \\ \Re \left (\frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{4}}\right ) \\ \sin \left (\frac {x}{2}\right ) \\ \frac {\cos \left (\frac {x}{2}\right )}{2}-\frac {\sqrt {3}\, \sin \left (\frac {x}{2}\right )}{2} \\ -\frac {\cos \left (\frac {x}{2}\right ) \sqrt {3}}{2}+\frac {\sin \left (\frac {x}{2}\right )}{2} \\ \cos \left (\frac {x}{2}\right ) \end {array}\right ]+c_{4} {\mathrm e}^{-\frac {\sqrt {3}\, x}{2}}\cdot \left [\begin {array}{c} \Im \left (\frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{5}}\right ) \\ \Im \left (\frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (-\frac {\mathrm {I}}{2}-\frac {\sqrt {3}}{2}\right )^{4}}\right ) \\ \cos \left (\frac {x}{2}\right ) \\ -\frac {\cos \left (\frac {x}{2}\right ) \sqrt {3}}{2}-\frac {\sin \left (\frac {x}{2}\right )}{2} \\ \frac {\cos \left (\frac {x}{2}\right )}{2}+\frac {\sqrt {3}\, \sin \left (\frac {x}{2}\right )}{2} \\ -\sin \left (\frac {x}{2}\right ) \end {array}\right ]+c_{5} {\mathrm e}^{\frac {\sqrt {3}\, x}{2}}\cdot \left [\begin {array}{c} \Re \left (\frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{5}}\right ) \\ \Re \left (\frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{4}}\right ) \\ \sin \left (\frac {x}{2}\right ) \\ \frac {\cos \left (\frac {x}{2}\right )}{2}+\frac {\sqrt {3}\, \sin \left (\frac {x}{2}\right )}{2} \\ \frac {\cos \left (\frac {x}{2}\right ) \sqrt {3}}{2}+\frac {\sin \left (\frac {x}{2}\right )}{2} \\ \cos \left (\frac {x}{2}\right ) \end {array}\right ]+c_{6} {\mathrm e}^{\frac {\sqrt {3}\, x}{2}}\cdot \left [\begin {array}{c} \Im \left (\frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{5}}\right ) \\ \Im \left (\frac {\cos \left (\frac {x}{2}\right )-\mathrm {I} \sin \left (\frac {x}{2}\right )}{\left (\frac {\sqrt {3}}{2}-\frac {\mathrm {I}}{2}\right )^{4}}\right ) \\ \cos \left (\frac {x}{2}\right ) \\ \frac {\cos \left (\frac {x}{2}\right ) \sqrt {3}}{2}-\frac {\sin \left (\frac {x}{2}\right )}{2} \\ \frac {\cos \left (\frac {x}{2}\right )}{2}-\frac {\sqrt {3}\, \sin \left (\frac {x}{2}\right )}{2} \\ -\sin \left (\frac {x}{2}\right ) \end {array}\right ]+\left [\begin {array}{c} c_{1} \sin \left (x \right )+c_{2} \cos \left (x \right ) \\ c_{1} \cos \left (x \right )-c_{2} \sin \left (x \right ) \\ -c_{1} \sin \left (x \right )-c_{2} \cos \left (x \right ) \\ -c_{1} \cos \left (x \right )+c_{2} \sin \left (x \right ) \\ c_{1} \sin \left (x \right )+c_{2} \cos \left (x \right ) \\ c_{1} \cos \left (x \right )-c_{2} \sin \left (x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=-32 c_{3} {\mathrm e}^{-\frac {\sqrt {3}\, x}{2}} \Im \left (\frac {\sin \left (\frac {x}{2}\right )+\mathrm {I} \cos \left (\frac {x}{2}\right )}{\left (\sqrt {3}+\mathrm {I}\right )^{5}}\right )+32 c_{4} {\mathrm e}^{-\frac {\sqrt {3}\, x}{2}} \Re \left (\frac {\sin \left (\frac {x}{2}\right )+\mathrm {I} \cos \left (\frac {x}{2}\right )}{\left (\sqrt {3}+\mathrm {I}\right )^{5}}\right )-32 c_{5} {\mathrm e}^{\frac {\sqrt {3}\, x}{2}} \Im \left (\frac {\sin \left (\frac {x}{2}\right )+\mathrm {I} \cos \left (\frac {x}{2}\right )}{\left (\mathrm {I}-\sqrt {3}\right )^{5}}\right )+32 c_{6} {\mathrm e}^{\frac {\sqrt {3}\, x}{2}} \Re \left (\frac {\sin \left (\frac {x}{2}\right )+\mathrm {I} \cos \left (\frac {x}{2}\right )}{\left (\mathrm {I}-\sqrt {3}\right )^{5}}\right )+c_{2} \cos \left (x \right )+c_{1} \sin \left (x \right ) \end {array} \]

Maple trace

`Methods for high order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
checking if the LODE has constant coefficients 
<- constant coefficients successful`
 

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 56

dsolve(diff(y(x),x$6)+y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = \left (-\sin \left (\frac {x}{2}\right ) c_{4} +c_{6} \cos \left (\frac {x}{2}\right )\right ) {\mathrm e}^{-\frac {\sqrt {3}\, x}{2}}+\left (\sin \left (\frac {x}{2}\right ) c_{3} +\cos \left (\frac {x}{2}\right ) c_{5} \right ) {\mathrm e}^{\frac {\sqrt {3}\, x}{2}}+c_{1} \sin \left (x \right )+c_{2} \cos \left (x \right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 92

DSolve[y''''''[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-\frac {\sqrt {3} x}{2}} \left (c_1 e^{\sqrt {3} x}+c_3\right ) \cos \left (\frac {x}{2}\right )+c_2 \cos (x)+c_4 e^{-\frac {\sqrt {3} x}{2}} \sin \left (\frac {x}{2}\right )+c_6 e^{\frac {\sqrt {3} x}{2}} \sin \left (\frac {x}{2}\right )+c_5 \sin (x) \]