80.11.20 problem 18
Internal
problem
ID
[21428]
Book
:
A
Textbook
on
Ordinary
Differential
Equations
by
Shair
Ahmad
and
Antonio
Ambrosetti.
Second
edition.
ISBN
978-3-319-16407-6.
Springer
2015
Section
:
Chapter
12.
Stability
theory.
Excercise
12.6
at
page
270
Problem
number
:
18
Date
solved
:
Thursday, October 02, 2025 at 07:31:27 PM
CAS
classification
:
[[_high_order, _missing_x]]
\begin{align*} x^{\prime \prime \prime \prime }+8 x^{\prime \prime \prime }+23 x^{\prime \prime }+2 x^{\prime }+12 x&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 37
ode:=diff(diff(diff(diff(x(t),t),t),t),t)+8*diff(diff(diff(x(t),t),t),t)+23*diff(diff(x(t),t),t)+2*diff(x(t),t)+12*x(t) = 0;
dsolve(ode,x(t), singsol=all);
\[
x = \moverset {4}{\munderset {\textit {\_a} =1}{\sum }}{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+8 \textit {\_Z}^{3}+23 \textit {\_Z}^{2}+2 \textit {\_Z} +12, \operatorname {index} =\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}
\]
✓ Mathematica. Time used: 0.002 (sec). Leaf size: 138
ode=D[x[t],{t,4}]+8*D[x[t],{t,3}]+23*D[x[t],{t,2}]+2*D[x[t],t]+12*x[t]==0;
ic={};
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to c_3 \exp \left (t \text {Root}\left [\text {$\#$1}^4+8 \text {$\#$1}^3+23 \text {$\#$1}^2+2 \text {$\#$1}+12\&,3\right ]\right )+c_4 \exp \left (t \text {Root}\left [\text {$\#$1}^4+8 \text {$\#$1}^3+23 \text {$\#$1}^2+2 \text {$\#$1}+12\&,4\right ]\right )+c_1 \exp \left (t \text {Root}\left [\text {$\#$1}^4+8 \text {$\#$1}^3+23 \text {$\#$1}^2+2 \text {$\#$1}+12\&,1\right ]\right )+c_2 \exp \left (t \text {Root}\left [\text {$\#$1}^4+8 \text {$\#$1}^3+23 \text {$\#$1}^2+2 \text {$\#$1}+12\&,2\right ]\right ) \end{align*}
✓ Sympy. Time used: 1.468 (sec). Leaf size: 536
from sympy import *
t = symbols("t")
x = Function("x")
ode = Eq(12*x(t) + 2*Derivative(x(t), t) + 23*Derivative(x(t), (t, 2)) + 8*Derivative(x(t), (t, 3)) + Derivative(x(t), (t, 4)),0)
ics = {}
dsolve(ode,func=x(t),ics=ics)
\[
x{\left (t \right )} = C_{1} e^{t \left (-2 + \frac {\sqrt {3} \sqrt {2 + \frac {625}{\sqrt [3]{10997 + 6 \sqrt {3422406} i}} + \sqrt [3]{10997 + 6 \sqrt {3422406} i}}}{6}\right )} \sin {\left (\frac {t \sqrt {\left |{\frac {4}{3} - \frac {\sqrt [3]{10997 + 6 \sqrt {3422406} i}}{3} + \frac {52 \sqrt {3}}{\sqrt {2 + \frac {625}{\sqrt [3]{10997 + 6 \sqrt {3422406} i}} + \sqrt [3]{10997 + 6 \sqrt {3422406} i}}} - \frac {625}{3 \sqrt [3]{10997 + 6 \sqrt {3422406} i}}}\right |}}{2} \right )} + C_{2} e^{t \left (-2 + \frac {\sqrt {3} \sqrt {2 + \frac {625}{\sqrt [3]{10997 + 6 \sqrt {3422406} i}} + \sqrt [3]{10997 + 6 \sqrt {3422406} i}}}{6}\right )} \cos {\left (\frac {t \sqrt {- \frac {4}{3} + \frac {625}{3 \sqrt [3]{10997 + 6 \sqrt {3422406} i}} - \frac {52 \sqrt {3}}{\sqrt {2 + \frac {625}{\sqrt [3]{10997 + 6 \sqrt {3422406} i}} + \sqrt [3]{10997 + 6 \sqrt {3422406} i}}} + \frac {\sqrt [3]{10997 + 6 \sqrt {3422406} i}}{3}}}{2} \right )} + C_{3} e^{- t \left (2 + \frac {\sqrt {3} \sqrt {2 + \frac {625}{\sqrt [3]{10997 + 6 \sqrt {3422406} i}} + \sqrt [3]{10997 + 6 \sqrt {3422406} i}}}{6}\right )} \sin {\left (\frac {t \sqrt {\left |{- \frac {4}{3} + \frac {625}{3 \sqrt [3]{10997 + 6 \sqrt {3422406} i}} + \frac {52 \sqrt {3}}{\sqrt {2 + \frac {625}{\sqrt [3]{10997 + 6 \sqrt {3422406} i}} + \sqrt [3]{10997 + 6 \sqrt {3422406} i}}} + \frac {\sqrt [3]{10997 + 6 \sqrt {3422406} i}}{3}}\right |}}{2} \right )} + C_{4} e^{- t \left (2 + \frac {\sqrt {3} \sqrt {2 + \frac {625}{\sqrt [3]{10997 + 6 \sqrt {3422406} i}} + \sqrt [3]{10997 + 6 \sqrt {3422406} i}}}{6}\right )} \cos {\left (\frac {t \sqrt {- \frac {4}{3} + \frac {625}{3 \sqrt [3]{10997 + 6 \sqrt {3422406} i}} + \frac {52 \sqrt {3}}{\sqrt {2 + \frac {625}{\sqrt [3]{10997 + 6 \sqrt {3422406} i}} + \sqrt [3]{10997 + 6 \sqrt {3422406} i}}} + \frac {\sqrt [3]{10997 + 6 \sqrt {3422406} i}}{3}}}{2} \right )}
\]