problem 63 (c)

Internal problem ID [14673]
Internal ﬁle name [OUTPUT/14353_Wednesday_April_03_2024_02_17_27_PM_24619359/index.tex]

Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section: Chapter 4. Higher Order Equations. Exercises 4.5, page 175
Problem number: 63 (c).
ODE order: 3.
ODE degree: 1.

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

\[ \boxed {\frac {31 y^{\prime \prime \prime }}{100}+\frac {56 y^{\prime \prime }}{5}-\frac {49 y^{\prime }}{5}+\frac {53 y}{10}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = -1, y^{\prime \prime }\left (0\right ) = 0] \end {align*}

The characteristic equation is \[ -\frac {49}{5} \lambda +\frac {56}{5} \lambda ^{2}+\frac {31}{100} \lambda ^{3}+\frac {53}{10} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= -\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}-\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}\\ \lambda _2 &= \frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{186}+\frac {672770}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}+\frac {i \sqrt {3}\, \left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}+\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}\right )}{2}\\ \lambda _3 &= \frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{186}+\frac {672770}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}-\frac {i \sqrt {3}\, \left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}+\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}\right )}{2} \end {align*}

Therefore the homogeneous solution is \[ y_h(t)={\mathrm e}^{\left (\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{186}+\frac {672770}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}+\frac {i \sqrt {3}\, \left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}+\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{1} +{\mathrm e}^{\left (\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{186}+\frac {672770}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}-\frac {i \sqrt {3}\, \left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}+\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{2} +{\mathrm e}^{\left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}-\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}\right ) t} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin {align*} y_1 &= {\mathrm e}^{\left (\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{186}+\frac {672770}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}+\frac {i \sqrt {3}\, \left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}+\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}\\ y_2 &= {\mathrm e}^{\left (\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{186}+\frac {672770}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}-\frac {i \sqrt {3}\, \left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}+\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}\right )}{2}\right ) t}\\ y_3 &= {\mathrm e}^{\left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}-\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}\right ) t} \end {align*}

Looking at the above solution \begin {align*} y = {\mathrm e}^{\left (\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{186}+\frac {672770}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}+\frac {i \sqrt {3}\, \left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}+\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{1} +{\mathrm e}^{\left (\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{186}+\frac {672770}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}-\frac {i \sqrt {3}\, \left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}+\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{2} +{\mathrm e}^{\left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}-\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}\right ) t} c_{3} \tag {1} \end {align*}

Initial conditions are now substituted in the above solution. This will generate the required equations to solve for the integration constants. substituting $y = -1$ and $t = 0$ in the above gives \begin {align*} -1 = c_{1} +c_{2} +c_{3}\tag {1A} \end {align*}

Taking derivative of the solution gives \begin {align*} y^{\prime } = \left (\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{186}+\frac {672770}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}+\frac {i \sqrt {3}\, \left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}+\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}\right )}{2}\right ) {\mathrm e}^{\left (\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{186}+\frac {672770}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}+\frac {i \sqrt {3}\, \left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}+\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{1} +\left (\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{186}+\frac {672770}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}-\frac {i \sqrt {3}\, \left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}+\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}\right )}{2}\right ) {\mathrm e}^{\left (\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{186}+\frac {672770}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}-\frac {i \sqrt {3}\, \left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}+\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{2} +\left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}-\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}\right ) {\mathrm e}^{\left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}-\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}\right ) t} c_{3} \end {align*}

substituting $y^{\prime } = -1$ and $t = 0$ in the above gives \begin {align*} -1 = \frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}} \left (-i \left (c_{1} -c_{2} \right ) \sqrt {3}+c_{1} +c_{2} -2 c_{3} \right )-2240 \left (c_{1} +c_{2} +c_{3} \right ) \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}-1345540 i \left (-c_{1} +c_{2} \right ) \sqrt {3}+1345540 c_{1} +1345540 c_{2} -2691080 c_{3}}{186 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}\tag {2A} \end {align*}

Taking two derivatives of the solution gives \begin {align*} y^{\prime \prime } = \left (\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{186}+\frac {672770}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}+\frac {i \sqrt {3}\, \left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}+\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2} {\mathrm e}^{\left (\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{186}+\frac {672770}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}+\frac {i \sqrt {3}\, \left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}+\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{1} +\left (\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{186}+\frac {672770}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}-\frac {i \sqrt {3}\, \left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}+\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2} {\mathrm e}^{\left (\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{186}+\frac {672770}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}-\frac {i \sqrt {3}\, \left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}+\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}\right )}{2}\right ) t} c_{2} +\left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}-\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}\right )^{2} {\mathrm e}^{\left (-\frac {\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{93}-\frac {1345540}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}-\frac {1120}{93}\right ) t} c_{3} \end {align*}

substituting $y^{\prime \prime } = 0$ and $t = 0$ in the above gives \begin {align*} 0 = \frac {\frac {1526309585 \left (-\frac {31 \left (c_{1} +c_{2} -2 c_{3} \right ) \sqrt {59667195849}}{305261917}-\frac {93 i \left (c_{1} -c_{2} \right ) \sqrt {19889065283}}{305261917}-i \left (c_{1} -c_{2} \right ) \sqrt {3}-c_{1} -c_{2} +2 c_{3} \right ) \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{5766}+\frac {1315160 \left (c_{1} +c_{2} +c_{3} \right ) \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}}}{2883}+\frac {5600 \left (-c_{1} -c_{2} +2 c_{3} \right ) \sqrt {59667195849}}{93}+\frac {5600 i \left (c_{1} -c_{2} \right ) \sqrt {19889065283}}{31}+\frac {885982799800 i \left (c_{1} -c_{2} \right ) \sqrt {3}}{2883}-\frac {885982799800 c_{1}}{2883}-\frac {885982799800 c_{2}}{2883}+\frac {1771965599600 c_{3}}{2883}}{\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}}}\tag {3A} \end {align*}

Equations {1A,2A,3A} are now solved for $\{c_{1}, c_{2}, c_{3}\}$. Solving for the constants gives \begin {align*} c_{1}&=\frac {\left (23673731757298 i \sqrt {3}\, \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {2}{3}}+163214002 i \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {2}{3}} \sqrt {19889065283}-154251220 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {2}{3}} \sqrt {19889065283}\, \sqrt {3}-2243665565509947 i \sqrt {3}\, \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}-379649387003 i \sqrt {19889065283}\, \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}-16362628041 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}} \sqrt {19889065283}\, \sqrt {3}-29581876331229344080 i \sqrt {3}+164845850396080 i \sqrt {19889065283}+225806225843800 \sqrt {19889065283}\, \sqrt {3}+19974365614660 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {2}{3}}-55067075370969947 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}+34559521450794478600\right ) \sqrt {3}}{6 \left (\left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {2}{3}}-1345540\right ) \left (1849683071319 \sqrt {3}+312983831 \sqrt {19889065283}\right )}\\ c_{2}&=\frac {140207675 i \sqrt {3}\, \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}} \sqrt {59667195849}}{308646111252519405624}+\frac {175942237 i \sqrt {59667195849}\, \sqrt {3}\, \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{321138394810653840}+\frac {4387 i \sqrt {3}\, \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}}}{77591909640}-\frac {1213 i \sqrt {3}\, \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{16146480}-\frac {140207675 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}} \sqrt {59667195849}}{308646111252519405624}+\frac {175942237 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}} \sqrt {59667195849}}{321138394810653840}-\frac {4387 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}}}{77591909640}-\frac {1213 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}{16146480}-\frac {1}{3}\\ c_{3}&=\frac {1228360 \sqrt {19889065283}\, \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}+175942237 \sqrt {3}\, \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {2}{3}}+1213 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {2}{3}} \sqrt {19889065283}-196290745000 \sqrt {3}\, \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}-18496830713190 \sqrt {3}-3129838310 \sqrt {19889065283}}{55490492139570 \sqrt {3}+9389514930 \sqrt {19889065283}} \end {align*}

Substituting these values back in above solution results in \begin {align*} y = \text {Expression too large to display} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\left (\left (19974365614660 \sqrt {3}-71021195271894 i-462753660 \sqrt {19889065283}-163214002 i \sqrt {59667195849}\right ) \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}}+\left (379649387003 i \sqrt {59667195849}+6730996696529841 i-55067075370969947 \sqrt {3}-49087884123 \sqrt {19889065283}\right ) \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}-164845850396080 i \sqrt {59667195849}+88745628993688032240 i+34559521450794478600 \sqrt {3}+677418677531400 \sqrt {19889065283}\right ) {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {2}{3}}-1345540 i \sqrt {3}+\left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {2}{3}}-2240 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}+1345540\right ) t}{186 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}}}+\left (\left (163214002 i \sqrt {59667195849}+19974365614660 \sqrt {3}-462753660 \sqrt {19889065283}+71021195271894 i\right ) \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}}+\left (-6730996696529841 i-55067075370969947 \sqrt {3}-49087884123 \sqrt {19889065283}-379649387003 i \sqrt {59667195849}\right ) \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}+164845850396080 i \sqrt {59667195849}-88745628993688032240 i+34559521450794478600 \sqrt {3}+677418677531400 \sqrt {19889065283}\right ) {\mathrm e}^{-\frac {t \left (\left (i \sqrt {3}-1\right ) \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {2}{3}}-1345540 i \sqrt {3}+2240 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}-1345540\right )}{186 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}}}+\left (\left (-51046829657234 \sqrt {3}-952395666 \sqrt {19889065283}\right ) \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}}+\left (110134150741939894 \sqrt {3}+98175768246 \sqrt {19889065283}\right ) \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}-54186107542893553640 \sqrt {3}+1171956228719640 \sqrt {19889065283}\right ) {\mathrm e}^{-\frac {\left (\left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {2}{3}}+1120 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}+1345540\right ) t}{93 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}}}}{6 \left (\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}}-1345540\right ) \left (1849683071319 \sqrt {3}+312983831 \sqrt {19889065283}\right )} \\ \end{align*}

Veriﬁcation of solutions

\[ y = \frac {\left (\left (19974365614660 \sqrt {3}-71021195271894 i-462753660 \sqrt {19889065283}-163214002 i \sqrt {59667195849}\right ) \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}}+\left (379649387003 i \sqrt {59667195849}+6730996696529841 i-55067075370969947 \sqrt {3}-49087884123 \sqrt {19889065283}\right ) \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}-164845850396080 i \sqrt {59667195849}+88745628993688032240 i+34559521450794478600 \sqrt {3}+677418677531400 \sqrt {19889065283}\right ) {\mathrm e}^{\frac {\left (i \sqrt {3}\, \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {2}{3}}-1345540 i \sqrt {3}+\left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {2}{3}}-2240 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}+1345540\right ) t}{186 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}}}+\left (\left (163214002 i \sqrt {59667195849}+19974365614660 \sqrt {3}-462753660 \sqrt {19889065283}+71021195271894 i\right ) \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}}+\left (-6730996696529841 i-55067075370969947 \sqrt {3}-49087884123 \sqrt {19889065283}-379649387003 i \sqrt {59667195849}\right ) \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}+164845850396080 i \sqrt {59667195849}-88745628993688032240 i+34559521450794478600 \sqrt {3}+677418677531400 \sqrt {19889065283}\right ) {\mathrm e}^{-\frac {t \left (\left (i \sqrt {3}-1\right ) \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {2}{3}}-1345540 i \sqrt {3}+2240 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}-1345540\right )}{186 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}}}+\left (\left (-51046829657234 \sqrt {3}-952395666 \sqrt {19889065283}\right ) \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}}+\left (110134150741939894 \sqrt {3}+98175768246 \sqrt {19889065283}\right ) \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}-54186107542893553640 \sqrt {3}+1171956228719640 \sqrt {19889065283}\right ) {\mathrm e}^{-\frac {\left (\left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {2}{3}}+1120 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}+1345540\right ) t}{93 \left (1564919155+465 \sqrt {19889065283}\, \sqrt {3}\right )^{\frac {1}{3}}}}}{6 \left (\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}}-1345540\right ) \left (1849683071319 \sqrt {3}+312983831 \sqrt {19889065283}\right )} \] Veriﬁed OK.

Maple trace

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

\[ y \left (t \right ) = \frac {\left (\left (\left (-1228360 \sqrt {3}\, \sqrt {19889065283}+588872235000\right ) \left (1564919155+465 \sqrt {3}\, \sqrt {19889065283}\right )^{\frac {1}{3}}+\left (-1213 \sqrt {3}\, \sqrt {19889065283}-527826711\right ) \left (1564919155+465 \sqrt {3}\, \sqrt {19889065283}\right )^{\frac {2}{3}}-6259676620 \sqrt {3}\, \sqrt {19889065283}-110980984279140\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (1564919155+465 \sqrt {3}\, \sqrt {19889065283}\right )^{\frac {2}{3}}-1345540\right ) t}{186 \left (1564919155+465 \sqrt {3}\, \sqrt {19889065283}\right )^{\frac {1}{3}}}\right )-527826711 \sin \left (\frac {\sqrt {3}\, \left (\left (1564919155+465 \sqrt {3}\, \sqrt {19889065283}\right )^{\frac {2}{3}}-1345540\right ) t}{186 \left (1564919155+465 \sqrt {3}\, \sqrt {19889065283}\right )^{\frac {1}{3}}}\right ) \left (\left (\frac {196290745000 \sqrt {3}}{175942237}-\frac {1228360 \sqrt {19889065283}}{175942237}\right ) \left (1564919155+465 \sqrt {3}\, \sqrt {19889065283}\right )^{\frac {1}{3}}+\left (1564919155+465 \sqrt {3}\, \sqrt {19889065283}\right )^{\frac {2}{3}} \left (\sqrt {3}+\frac {1213 \sqrt {19889065283}}{175942237}\right )\right )\right ) {\mathrm e}^{\frac {\left (1345540+\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}}-2240 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}\right ) t}{186 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}}+1213 \,{\mathrm e}^{-\frac {\left (\left (1564919155+465 \sqrt {59667195849}\right )^{\frac {2}{3}}+1120 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}+1345540\right ) t}{93 \left (1564919155+465 \sqrt {59667195849}\right )^{\frac {1}{3}}}} \left (\left (\frac {1228360 \sqrt {3}\, \sqrt {19889065283}}{1213}-\frac {588872235000}{1213}\right ) \left (1564919155+465 \sqrt {3}\, \sqrt {19889065283}\right )^{\frac {1}{3}}+\left (\sqrt {3}\, \sqrt {19889065283}+\frac {527826711}{1213}\right ) \left (1564919155+465 \sqrt {3}\, \sqrt {19889065283}\right )^{\frac {2}{3}}-\frac {3129838310 \sqrt {3}\, \sqrt {19889065283}}{1213}-\frac {55490492139570}{1213}\right )}{9389514930 \sqrt {3}\, \sqrt {19889065283}+166471476418710} \]

\[ y(t)\to -\frac {e^{t \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,2\right ]} \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,1\right ]^2-e^{t \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,3\right ]} \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,1\right ]^2+e^{t \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,3\right ]} \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,1\right ]^2 \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,2\right ]-e^{t \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,1\right ]} \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,2\right ]^2+e^{t \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,3\right ]} \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,2\right ]^2-e^{t \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,3\right ]} \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,1\right ] \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,2\right ]^2-e^{t \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,2\right ]} \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,1\right ]^2 \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,3\right ]+e^{t \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,1\right ]} \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,2\right ]^2 \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,3\right ]+e^{t \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,1\right ]} \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,3\right ]^2-e^{t \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,2\right ]} \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,3\right ]^2+e^{t \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,2\right ]} \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,1\right ] \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,3\right ]^2-e^{t \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,1\right ]} \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,2\right ] \text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,3\right ]^2}{\left (\text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,1\right ]-\text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,2\right ]\right ) \left (-\text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,1\right ]+\text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,3\right ]\right ) \left (-\text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,2\right ]+\text {Root}\left [31 \text {$\#$1}^3+1120 \text {$\#$1}^2-980 \text {$\#$1}+530\&,3\right ]\right )} \]

13.38 problem 63 (c)