80.11.14 problem 14 (a)
Internal
problem
ID
[21422]
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
:
14
(a)
Date
solved
:
Thursday, October 02, 2025 at 07:31:07 PM
CAS
classification
:
[[_3rd_order, _missing_x]]
\begin{align*} x^{\prime \prime \prime }+a x^{\prime \prime }+b x^{\prime }+c x&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 490
ode:=diff(diff(diff(x(t),t),t),t)+a*diff(diff(x(t),t),t)+b*diff(x(t),t)+c*x(t) = 0;
dsolve(ode,x(t), singsol=all);
\[
x = {\mathrm e}^{-\frac {t a}{3}} \left (c_3 \,{\mathrm e}^{\frac {t \left (\left (36 b a -108 c -8 a^{3}+12 \sqrt {81 c^{2}+\left (12 a^{3}-54 b a \right ) c -3 b^{2} a^{2}+12 b^{3}}\right )^{{2}/{3}}+4 a^{2}-12 b \right )}{6 \left (36 b a -108 c -8 a^{3}+12 \sqrt {81 c^{2}+\left (12 a^{3}-54 b a \right ) c -3 b^{2} a^{2}+12 b^{3}}\right )^{{1}/{3}}}}+c_2 \,{\mathrm e}^{\frac {\left (i \left (\left (36 b a -108 c -8 a^{3}+12 \sqrt {81 c^{2}+\left (12 a^{3}-54 b a \right ) c -3 b^{2} a^{2}+12 b^{3}}\right )^{{2}/{3}}-4 a^{2}+12 b \right ) \sqrt {3}-\left (36 b a -108 c -8 a^{3}+12 \sqrt {81 c^{2}+\left (12 a^{3}-54 b a \right ) c -3 b^{2} a^{2}+12 b^{3}}\right )^{{2}/{3}}-4 a^{2}+12 b \right ) t}{12 \left (36 b a -108 c -8 a^{3}+12 \sqrt {81 c^{2}+\left (12 a^{3}-54 b a \right ) c -3 b^{2} a^{2}+12 b^{3}}\right )^{{1}/{3}}}}+c_1 \,{\mathrm e}^{\frac {\left (-i \left (\left (36 b a -108 c -8 a^{3}+12 \sqrt {81 c^{2}+\left (12 a^{3}-54 b a \right ) c -3 b^{2} a^{2}+12 b^{3}}\right )^{{2}/{3}}-4 a^{2}+12 b \right ) \sqrt {3}-\left (36 b a -108 c -8 a^{3}+12 \sqrt {81 c^{2}+\left (12 a^{3}-54 b a \right ) c -3 b^{2} a^{2}+12 b^{3}}\right )^{{2}/{3}}-4 a^{2}+12 b \right ) t}{12 \left (36 b a -108 c -8 a^{3}+12 \sqrt {81 c^{2}+\left (12 a^{3}-54 b a \right ) c -3 b^{2} a^{2}+12 b^{3}}\right )^{{1}/{3}}}}\right )
\]
✓ Mathematica. Time used: 0.004 (sec). Leaf size: 84
ode=D[x[t],{t,3}]+a*D[x[t],{t,2}]+b*D[x[t],t]+c*x[t]==0;
ic={};
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to c_1 e^{t \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2 a+\text {$\#$1} b+c\&,1\right ]}+c_2 e^{t \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2 a+\text {$\#$1} b+c\&,2\right ]}+c_3 e^{t \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2 a+\text {$\#$1} b+c\&,3\right ]} \end{align*}
✓ Sympy. Time used: 0.205 (sec). Leaf size: 24
from sympy import *
t = symbols("t")
a = symbols("a")
b = symbols("b")
c = symbols("c")
x = Function("x")
ode = Eq(a*Derivative(x1(t), (t, 2)) + b*Derivative(x1(t), t) + c*x(t) + Derivative(x1(t), (t, 3)),0)
ics = {}
dsolve(ode,func=x(t),ics=ics)
\[
x{\left (t \right )} = \frac {- a \frac {d^{2}}{d t^{2}} x_{1}{\left (t \right )} - b \frac {d}{d t} x_{1}{\left (t \right )} - \frac {d^{3}}{d t^{3}} x_{1}{\left (t \right )}}{c}
\]