5.15 problem 26

Internal problem ID [4805]
Internal file name [OUTPUT/4298_Sunday_June_05_2022_12_56_25_PM_92400865/index.tex]

Book: Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section: Chapter 8, Ordinary differential equations. Section 5. SECOND-ORDER LINEAR EQUATIONSWITH CONSTANT COEFFICIENTS AND ZERO RIGHT-HAND SIDE. page 414
Problem number: 26.
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]]

\[ \boxed {y^{\prime \prime \prime }+3 y^{\prime \prime }-9 y^{\prime }-5 y=0} \] The characteristic equation is \[ \lambda ^{3}+3 \lambda ^{2}-9 \lambda -5 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= \left (-3+i \sqrt {55}\right )^{\frac {1}{3}}+\frac {4}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}-1\\ \lambda _2 &= -\frac {\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}-1+\frac {i \sqrt {3}\, \left (\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\\ \lambda _3 &= -\frac {\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}-1-\frac {i \sqrt {3}\, \left (\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}

Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{\left (\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}+\frac {4}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}-1\right ) x} c_{1} +{\mathrm e}^{\left (-\frac {\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}-1+\frac {i \sqrt {3}\, \left (\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}-1-\frac {i \sqrt {3}\, \left (\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{\left (\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}+\frac {4}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}-1\right ) x}\\ y_2 &= {\mathrm e}^{\left (-\frac {\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}-1+\frac {i \sqrt {3}\, \left (\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\right ) x}\\ y_3 &= {\mathrm e}^{\left (-\frac {\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}-1-\frac {i \sqrt {3}\, \left (\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\left (\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}+\frac {4}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}-1\right ) x} c_{1} +{\mathrm e}^{\left (-\frac {\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}-1+\frac {i \sqrt {3}\, \left (\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}-1-\frac {i \sqrt {3}\, \left (\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{3} \\ \end{align*}

Verification of solutions

\[ y = {\mathrm e}^{\left (\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}+\frac {4}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}-1\right ) x} c_{1} +{\mathrm e}^{\left (-\frac {\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}-1+\frac {i \sqrt {3}\, \left (\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}-1-\frac {i \sqrt {3}\, \left (\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{3} \] Verified OK.

5.15.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }+3 y^{\prime \prime }-9 y^{\prime }-5 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 )=-3 y_{3}\left (x \right )+9 y_{2}\left (x \right )+5 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 )=-3 y_{3}\left (x \right )+9 y_{2}\left (x \right )+5 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 \\ 5 & 9 & -3 \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 \\ 5 & 9 & -3 \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 [\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}+\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1, \left [\begin {array}{c} \frac {1}{\left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}+\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1\right )^{2}} \\ \frac {1}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}+\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1} \\ 1 \end {array}\right ]\right ], \left [-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1-\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1-\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1-\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ]\right ], \left [-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1+\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1+\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1+\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}+\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1, \left [\begin {array}{c} \frac {1}{\left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}+\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1\right )^{2}} \\ \frac {1}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}+\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}={\mathrm e}^{\left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}+\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}+\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1\right )^{2}} \\ \frac {1}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}+\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1-\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1-\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1-\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{2}={\mathrm e}^{\left (-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1-\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1-\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1-\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1+\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1+\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1+\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{3}={\mathrm e}^{\left (-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1+\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1+\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1+\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\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}+c_{3} {\moverset {\rightarrow }{y}}_{3} \\ \bullet & {} & \textrm {Substitute solutions into the general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}={\mathrm e}^{\left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}+\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1\right ) x} c_{1} \cdot \left [\begin {array}{c} \frac {1}{\left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}+\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1\right )^{2}} \\ \frac {1}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}+\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1} \\ 1 \end {array}\right ]+c_{2} {\mathrm e}^{\left (-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1-\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1-\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1-\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ]+c_{3} {\mathrm e}^{\left (-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1+\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1+\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}} \\ \frac {1}{-\frac {\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}-1+\frac {\mathrm {I} \sqrt {3}\, \left (\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {37 \left (c_{2} {\mathrm e}^{\left (-1-2 \sin \left (\frac {\arctan \left (\frac {\sqrt {55}}{3}\right )}{3}+\frac {\pi }{6}\right )+2 \sqrt {3}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {55}}{3}\right )}{3}+\frac {\pi }{6}\right )\right ) x} \left (\frac {231 \,\mathrm {I}}{37}+\frac {2 \left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {2}{3}} \left (-33 \,\mathrm {I}+\mathrm {I} \sqrt {165}+\sqrt {55}+33 \sqrt {3}\right )}{37}+\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}} \left (\sqrt {55}+3 \,\mathrm {I}\right )-\frac {19 \sqrt {55}}{37}+\frac {231 \sqrt {3}}{37}+\frac {19 \,\mathrm {I} \sqrt {165}}{37}\right )+\left (\frac {231 \,\mathrm {I}}{37}+\frac {2 \left (-33 \,\mathrm {I}-\mathrm {I} \sqrt {165}-33 \sqrt {3}+\sqrt {55}\right ) \left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {2}{3}}}{37}+\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}} \left (\sqrt {55}+3 \,\mathrm {I}\right )-\frac {19 \sqrt {55}}{37}-\frac {231 \sqrt {3}}{37}-\frac {19 \,\mathrm {I} \sqrt {165}}{37}\right ) c_{3} {\mathrm e}^{-2 \left (\sqrt {3}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {55}}{3}\right )}{3}+\frac {\pi }{6}\right )+\sin \left (\frac {\arctan \left (\frac {\sqrt {55}}{3}\right )}{3}+\frac {\pi }{6}\right )+\frac {1}{2}\right ) x}+{\mathrm e}^{\left (4 \sin \left (\frac {\arctan \left (\frac {\sqrt {55}}{3}\right )}{3}+\frac {\pi }{6}\right )-1\right ) x} c_{1} \left (-\frac {462 \,\mathrm {I}}{37}+\frac {4 \left (33 \,\mathrm {I}-\sqrt {55}\right ) \left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {2}{3}}}{37}+\left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {1}{3}} \left (\sqrt {55}+3 \,\mathrm {I}\right )+\frac {38 \sqrt {55}}{37}\right )\right ) \left (-3+\mathrm {I} \sqrt {55}\right )^{\frac {2}{3}}}{50 \left (-3 \sqrt {55}+23 \,\mathrm {I}\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: 0.016 (sec). Leaf size: 105

dsolve(diff(y(x),x$3)+3*diff(y(x),x$2)-9*diff(y(x),x)-5*y(x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = c_{1} {\mathrm e}^{\left (-1-2 \sin \left (\frac {\arctan \left (\frac {\sqrt {55}}{3}\right )}{3}+\frac {\pi }{6}\right )+2 \sqrt {3}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {55}}{3}\right )}{3}+\frac {\pi }{6}\right )\right ) x}+c_{2} {\mathrm e}^{-2 \left (\sqrt {3}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {55}}{3}\right )}{3}+\frac {\pi }{6}\right )+\sin \left (\frac {\arctan \left (\frac {\sqrt {55}}{3}\right )}{3}+\frac {\pi }{6}\right )+\frac {1}{2}\right ) x}+c_{3} {\mathrm e}^{\left (4 \sin \left (\frac {\arctan \left (\frac {\sqrt {55}}{3}\right )}{3}+\frac {\pi }{6}\right )-1\right ) x} \]

✓ Solution by Mathematica

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

DSolve[y'''[x]+3*y''[x]-9*y'[x]-5*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-9 \text {$\#$1}-5\&,2\right ]\right )+c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-9 \text {$\#$1}-5\&,3\right ]\right )+c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2-9 \text {$\#$1}-5\&,1\right ]\right ) \]