63.14.4 problem 1(d)
Internal
problem
ID
[13109]
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
:
Monday, March 31, 2025 at 07:34:31 AM
CAS
classification
:
[[_3rd_order, _missing_x]]
\begin{align*} x^{\prime \prime \prime }-x^{\prime }-8 x&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 128
ode:=diff(diff(diff(x(t),t),t),t)-diff(x(t),t)-8*x(t) = 0;
dsolve(ode,x(t), singsol=all);
\[
x = c_1 \,{\mathrm e}^{\frac {\left (\left (108+3 \sqrt {1293}\right )^{{2}/{3}}+3\right ) t}{3 \left (108+3 \sqrt {1293}\right )^{{1}/{3}}}}+{\mathrm e}^{-\frac {\left (\left (108+3 \sqrt {1293}\right )^{{2}/{3}}+3\right ) t}{6 \left (108+3 \sqrt {1293}\right )^{{1}/{3}}}} \left (-c_2 \sin \left (\frac {\sqrt {3}\, \left (\left (108+3 \sqrt {3}\, \sqrt {431}\right )^{{2}/{3}}-3\right ) t}{6 \left (108+3 \sqrt {3}\, \sqrt {431}\right )^{{1}/{3}}}\right )+c_3 \cos \left (\frac {\sqrt {3}\, \left (\left (108+3 \sqrt {3}\, \sqrt {431}\right )^{{2}/{3}}-3\right ) t}{6 \left (108+3 \sqrt {3}\, \sqrt {431}\right )^{{1}/{3}}}\right )\right )
\]
✓ Mathematica. Time used: 0.002 (sec). Leaf size: 69
ode=D[x[t],{t,3}]-D[x[t],t]-8*x[t]==0;
ic={};
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
\[
x(t)\to c_2 \exp \left (t \text {Root}\left [\text {$\#$1}^3-\text {$\#$1}-8\&,2\right ]\right )+c_3 \exp \left (t \text {Root}\left [\text {$\#$1}^3-\text {$\#$1}-8\&,3\right ]\right )+c_1 \exp \left (t \text {Root}\left [\text {$\#$1}^3-\text {$\#$1}-8\&,1\right ]\right )
\]
✓ Sympy. Time used: 0.558 (sec). Leaf size: 177
from sympy import *
t = symbols("t")
x = Function("x")
ode = Eq(-8*x(t) - Derivative(x(t), t) + Derivative(x(t), (t, 3)),0)
ics = {}
dsolve(ode,func=x(t),ics=ics)
\[
x{\left (t \right )} = C_{1} e^{- \frac {t \left (\frac {1}{\sqrt [3]{\frac {\sqrt {1293}}{9} + 4}} + 3 \sqrt [3]{\frac {\sqrt {1293}}{9} + 4}\right )}{6}} \sin {\left (\frac {\sqrt {3} t \left (- 3 \sqrt [3]{\frac {\sqrt {1293}}{9} + 4} + \frac {1}{\sqrt [3]{\frac {\sqrt {1293}}{9} + 4}}\right )}{6} \right )} + C_{2} e^{- \frac {t \left (\frac {1}{\sqrt [3]{\frac {\sqrt {1293}}{9} + 4}} + 3 \sqrt [3]{\frac {\sqrt {1293}}{9} + 4}\right )}{6}} \cos {\left (\frac {\sqrt {3} t \left (- 3 \sqrt [3]{\frac {\sqrt {1293}}{9} + 4} + \frac {1}{\sqrt [3]{\frac {\sqrt {1293}}{9} + 4}}\right )}{6} \right )} + C_{3} e^{t \left (\frac {1}{3 \sqrt [3]{\frac {\sqrt {1293}}{9} + 4}} + \sqrt [3]{\frac {\sqrt {1293}}{9} + 4}\right )}
\]