1.7 problem 16

Internal problem ID [818]
Internal file name [OUTPUT/818_Sunday_June_05_2022_01_50_30_AM_97126028/index.tex]

Book: Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima, Meade
Section: Chapter 4.1, Higher order linear differential equations. General theory. page 173
Problem number: 16.
ODE order: 3.
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

[[_3rd_order, _missing_x]]

\[ \boxed {y^{\prime \prime \prime }+2 y^{\prime \prime }-y^{\prime }-3 y=0} \] The characteristic equation is \[ \lambda ^{3}+2 \lambda ^{2}-\lambda -3 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= \frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}\\ \lambda _2 &= -\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\\ \lambda _3 &= -\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}

Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{1} +{\mathrm e}^{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{2} +{\mathrm e}^{\left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}\right ) x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right ) x}\\ y_2 &= {\mathrm e}^{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right ) x}\\ y_3 &= {\mathrm e}^{\left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}\right ) x} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{1} +{\mathrm e}^{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{2} +{\mathrm e}^{\left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}\right ) x} c_{3} \\ \end{align*}

Verification of solutions

\[ y = {\mathrm e}^{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{1} +{\mathrm e}^{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{2} +{\mathrm e}^{\left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}\right ) x} c_{3} \] Verified OK.

1.7.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }+2 y^{\prime \prime }-y^{\prime }-3 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & y^{\prime \prime \prime } \\ \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 {Isolate for}\hspace {3pt} y_{3}^{\prime }\left (x \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & y_{3}^{\prime }\left (x \right )=-2 y_{3}\left (x \right )+y_{2}\left (x \right )+3 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_{3}^{\prime }\left (x \right )=-2 y_{3}\left (x \right )+y_{2}\left (x \right )+3 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 ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (x \right )=\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 3 & 1 & -2 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (x \right ) \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 3 & 1 & -2 \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 [\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}, \left [\begin {array}{c} \frac {1}{\left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}\right )^{2}} \\ \frac {1}{\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}} \\ 1 \end {array}\right ]\right ], \left [-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ]\right ], \left [-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}, \left [\begin {array}{c} \frac {1}{\left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}\right )^{2}} \\ \frac {1}{\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}={\mathrm e}^{\left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}\right )^{2}} \\ \frac {1}{\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Use Euler identity to write solution in terms of}\hspace {3pt} \sin \hspace {3pt}\textrm {and}\hspace {3pt} \cos \\ {} & {} & {\mathrm e}^{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}\right ) x}\cdot \left (\cos \left (\frac {\sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) x}{2}\right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}\right ) x}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {\sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) x}{2}\right )}{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {\cos \left (\frac {\sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) x}{2}\right )}{-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )}{2}} \\ \cos \left (\frac {\sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) 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}}_{2}\left (x \right )={\mathrm e}^{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}\right ) x}\cdot \left [\begin {array}{c} -\frac {18 \left (188+12 \sqrt {93}\right )^{\frac {2}{3}} \left (\left (188+12 \sqrt {93}\right )^{\frac {4}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+\left (188+12 \sqrt {93}\right )^{\frac {4}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )-144 \left (188+12 \sqrt {93}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )-224 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+96 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {93}-224 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )-96 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {93}+720 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}-720 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )\right )}{\left (\left (188+12 \sqrt {93}\right )^{\frac {4}{3}}+1536+48 \sqrt {93}-12 \left (188+12 \sqrt {93}\right )^{\frac {2}{3}}+112 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}\right )^{2}} \\ -\frac {3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}} \left (-\left (188+12 \sqrt {93}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+\left (188+12 \sqrt {93}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )+8 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )+28 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+28 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )\right )}{\left (188+12 \sqrt {93}\right )^{\frac {4}{3}}+1536+48 \sqrt {93}-12 \left (188+12 \sqrt {93}\right )^{\frac {2}{3}}+112 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}} \\ \cos \left (\frac {\sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) x}{2}\right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{3}\left (x \right )={\mathrm e}^{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}\right ) x}\cdot \left [\begin {array}{c} -\frac {18 \left (188+12 \sqrt {93}\right )^{\frac {2}{3}} \left (\left (188+12 \sqrt {93}\right )^{\frac {4}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}-\left (188+12 \sqrt {93}\right )^{\frac {4}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )+144 \left (188+12 \sqrt {93}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )-224 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+96 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {93}+224 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )+96 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {93}+720 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+720 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )\right )}{\left (\left (188+12 \sqrt {93}\right )^{\frac {4}{3}}+1536+48 \sqrt {93}-12 \left (188+12 \sqrt {93}\right )^{\frac {2}{3}}+112 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}\right )^{2}} \\ \frac {3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}} \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+\left (188+12 \sqrt {93}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )+8 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )-28 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+28 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )\right )}{\left (188+12 \sqrt {93}\right )^{\frac {4}{3}}+1536+48 \sqrt {93}-12 \left (188+12 \sqrt {93}\right )^{\frac {2}{3}}+112 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}} \\ -\sin \left (\frac {\sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) 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}+c_{2} {\moverset {\rightarrow }{y}}_{2}\left (x \right )+c_{3} {\moverset {\rightarrow }{y}}_{3}\left (x \right ) \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}=c_{1} {\mathrm e}^{\left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}\right )^{2}} \\ \frac {1}{\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}+\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}} \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}\right ) x}\cdot \left [\begin {array}{c} -\frac {18 \left (188+12 \sqrt {93}\right )^{\frac {2}{3}} \left (\left (188+12 \sqrt {93}\right )^{\frac {4}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+\left (188+12 \sqrt {93}\right )^{\frac {4}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )-144 \left (188+12 \sqrt {93}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )-224 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+96 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {93}-224 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )-96 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {93}+720 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}-720 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )\right )}{\left (\left (188+12 \sqrt {93}\right )^{\frac {4}{3}}+1536+48 \sqrt {93}-12 \left (188+12 \sqrt {93}\right )^{\frac {2}{3}}+112 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}\right )^{2}} \\ -\frac {3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}} \left (-\left (188+12 \sqrt {93}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+\left (188+12 \sqrt {93}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )+8 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )+28 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+28 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )\right )}{\left (188+12 \sqrt {93}\right )^{\frac {4}{3}}+1536+48 \sqrt {93}-12 \left (188+12 \sqrt {93}\right )^{\frac {2}{3}}+112 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}} \\ \cos \left (\frac {\sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) x}{2}\right ) \end {array}\right ]+c_{3} {\mathrm e}^{\left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{12}-\frac {7}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}-\frac {2}{3}\right ) x}\cdot \left [\begin {array}{c} -\frac {18 \left (188+12 \sqrt {93}\right )^{\frac {2}{3}} \left (\left (188+12 \sqrt {93}\right )^{\frac {4}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}-\left (188+12 \sqrt {93}\right )^{\frac {4}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )+144 \left (188+12 \sqrt {93}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )-224 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+96 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}\, \sqrt {93}+224 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )+96 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {93}+720 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+720 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )\right )}{\left (\left (188+12 \sqrt {93}\right )^{\frac {4}{3}}+1536+48 \sqrt {93}-12 \left (188+12 \sqrt {93}\right )^{\frac {2}{3}}+112 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}\right )^{2}} \\ \frac {3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}} \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+\left (188+12 \sqrt {93}\right )^{\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )+8 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )-28 \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+28 \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right )\right )}{\left (188+12 \sqrt {93}\right )^{\frac {4}{3}}+1536+48 \sqrt {93}-12 \left (188+12 \sqrt {93}\right )^{\frac {2}{3}}+112 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}} \\ -\sin \left (\frac {\sqrt {3}\, \left (\frac {\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}{6}-\frac {14}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}\right ) x}{2}\right ) \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {13 \left (\left (\frac {40}{3}+\frac {7 \left (\sqrt {3}\, \sqrt {31}+11\right ) \left (188+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}}{78}+\left (\frac {47}{3}+\sqrt {3}\, \sqrt {31}\right ) \left (188+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}+\frac {4 \sqrt {3}\, \sqrt {31}}{39}\right ) c_{1} {\mathrm e}^{-\frac {2 x \left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}}{4}+\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}-7\right )}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}}+\left (\left (\frac {7 \left (\left (-\frac {\sqrt {3}\, c_{2}}{3}-c_{3} \right ) \sqrt {31}-\frac {11 c_{2}}{3}-\frac {11 c_{3} \sqrt {3}}{3}\right ) \left (188+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}}{52}+c_{2} \left (\frac {47}{3}+\sqrt {3}\, \sqrt {31}\right ) \left (188+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}+\frac {2 \left (c_{3} -\frac {\sqrt {3}\, c_{2}}{3}\right ) \sqrt {31}}{13}+\frac {20 c_{3} \sqrt {3}}{3}-\frac {20 c_{2}}{3}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}\right )-\sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}\right ) \left (\frac {7 \left (\left (-\frac {c_{3} \sqrt {3}}{3}+c_{2} \right ) \sqrt {31}+\frac {11 \sqrt {3}\, c_{2}}{3}-\frac {11 c_{3}}{3}\right ) \left (188+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}}{52}+c_{3} \left (\frac {47}{3}+\sqrt {3}\, \sqrt {31}\right ) \left (188+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}+\frac {2 \left (-\frac {c_{3} \sqrt {3}}{3}-c_{2} \right ) \sqrt {31}}{13}-\frac {20 \sqrt {3}\, c_{2}}{3}-\frac {20 c_{3}}{3}\right )\right ) {\mathrm e}^{-\frac {\left (\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}+8 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}+28\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}}\right ) \left (188+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}}{72 \left (141 \sqrt {3}\, \sqrt {31}+1523\right )} \end {array} \]

Maple trace

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

✓ Solution by Maple

Time used: 0.0 (sec). Leaf size: 183

dsolve(diff(y(x),x$3)+2*diff(y(x),x$2)-diff(y(x),x)-3*y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {2 x \left (-\frac {\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}}{4}+\left (188+12 \sqrt {93}\right )^{\frac {1}{3}}-7\right )}{3 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}}-c_{2} {\mathrm e}^{-\frac {\left (28+\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}+8 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}\right )+c_{3} {\mathrm e}^{-\frac {\left (28+\left (188+12 \sqrt {93}\right )^{\frac {2}{3}}+8 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}\right ) x}{12 \left (188+12 \sqrt {93}\right )^{\frac {1}{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {2}{3}}-28\right ) x}{12 \left (188+12 \sqrt {3}\, \sqrt {31}\right )^{\frac {1}{3}}}\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 87

DSolve[y'''[x]+2*y''[x]-y'[x]-3*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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