63.14.4 problem 1(d)

Internal problem ID [13180]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 2, Second order linear equations. Section 2.5 Higher order equations. Exercises page 130
Problem number : 1(d)
Date solved : Tuesday, January 28, 2025 at 05:12:02 AM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} x^{\prime \prime \prime }-x^{\prime }-8 x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 150

dsolve(diff(x(t),t$3)-diff(x(t),t)-8*x(t)=0,x(t), singsol=all)
 

\[ x \left (t \right ) = c_{1} {\mathrm e}^{\frac {\left (\left (108+3 \sqrt {1293}\right )^{{2}/{3}}+3\right ) t}{3 \left (108+3 \sqrt {1293}\right )^{{1}/{3}}}}-c_{2} {\mathrm e}^{-\frac {\left (\left (108+3 \sqrt {1293}\right )^{{2}/{3}}+3\right ) t}{6 \left (108+3 \sqrt {1293}\right )^{{1}/{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (108+3 \sqrt {3}\, \sqrt {431}\right )^{{2}/{3}}-3\right ) t}{6 \left (108+3 \sqrt {3}\, \sqrt {431}\right )^{{1}/{3}}}\right )+c_{3} {\mathrm e}^{-\frac {\left (\left (108+3 \sqrt {1293}\right )^{{2}/{3}}+3\right ) t}{6 \left (108+3 \sqrt {1293}\right )^{{1}/{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (108+3 \sqrt {3}\, \sqrt {431}\right )^{{2}/{3}}-3\right ) t}{6 \left (108+3 \sqrt {3}\, \sqrt {431}\right )^{{1}/{3}}}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 69

DSolve[D[x[t],{t,3}]-D[x[t],t]-8*x[t]==0,x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to c_2 \exp \left (t \text {Root}\left [\text {$\#$1}^3-\text {$\#$1}-8\&,2\right ]\right )+c_3 \exp \left (t \text {Root}\left [\text {$\#$1}^3-\text {$\#$1}-8\&,3\right ]\right )+c_1 \exp \left (t \text {Root}\left [\text {$\#$1}^3-\text {$\#$1}-8\&,1\right ]\right ) \]