Internal problem ID [10486]
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).
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _missing_x]]
\[ \boxed {x^{\prime \prime \prime }-x^{\prime }-8 x=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 150
dsolve(diff(x(t),t$3)-diff(x(t),t)-8*x(t)=0,x(t), singsol=all)
\[ x \left (t \right ) = c_{1} {\mathrm e}^{\frac {\left (\left (108+3 \sqrt {1293}\right )^{\frac {2}{3}}+3\right ) t}{3 \left (108+3 \sqrt {1293}\right )^{\frac {1}{3}}}}-c_{2} {\mathrm e}^{\frac {\left (-\frac {\left (108+3 \sqrt {1293}\right )^{\frac {2}{3}}}{6}-\frac {1}{2}\right ) t}{\left (108+3 \sqrt {1293}\right )^{\frac {1}{3}}}} \sin \left (\frac {\left (\left (108+3 \sqrt {1293}\right )^{\frac {2}{3}} \sqrt {3}-3 \sqrt {3}\right ) t}{6 \left (108+3 \sqrt {1293}\right )^{\frac {1}{3}}}\right )+c_{3} {\mathrm e}^{\frac {\left (-\frac {\left (108+3 \sqrt {1293}\right )^{\frac {2}{3}}}{6}-\frac {1}{2}\right ) t}{\left (108+3 \sqrt {1293}\right )^{\frac {1}{3}}}} \cos \left (\frac {\left (\left (108+3 \sqrt {1293}\right )^{\frac {2}{3}} \sqrt {3}-3 \sqrt {3}\right ) t}{6 \left (108+3 \sqrt {1293}\right )^{\frac {1}{3}}}\right ) \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 69
DSolve[x'''[t]-x'[t]-8*x[t]==0,x[t],t,IncludeSingularSolutions -> True]
\begin{align*} 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 ) \\ \end{align*}