80.6.10 problem 10
Internal
problem
ID
[21300]
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
6.
Higher
order
linear
equations.
Excercise
6.5
at
page
133
Problem
number
:
10
Date
solved
:
Friday, October 03, 2025 at 07:49:28 AM
CAS
classification
:
[[_3rd_order, _missing_x]]
\begin{align*} x^{\prime \prime \prime }+a x^{\prime \prime }+b x^{\prime }+c x&=0 \end{align*}
With initial conditions
\begin{align*}
x \left (\infty \right )&=0 \\
\end{align*}
✓ Maple. Time used: 3.041 (sec). Leaf size: 561
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;
ic:=[x(infinity) = 0];
dsolve([ode,op(ic)],x(t), singsol=all);
\[
x = \munderset {\textit {\_a} \rightarrow \infty }{\operatorname {lim}}{\mathrm e}^{-\frac {t a}{3}} \left (-c_1 \,{\mathrm e}^{\frac {\left (-\frac {i \sqrt {3}\, \textit {\_a}}{4}-\frac {3 \textit {\_a}}{4}+\frac {t}{2}\right ) \left (36 b a -108 c -8 a^{3}+12 \sqrt {12 c \,a^{3}-3 b^{2} a^{2}-54 b a c +12 b^{3}+81 c^{2}}\right )^{{2}/{3}}+\left (i \sqrt {3}\, \textit {\_a} -3 \textit {\_a} +2 t \right ) \left (a^{2}-3 b \right )}{3 \left (36 b a -108 c -8 a^{3}+12 \sqrt {12 c \,a^{3}-3 b^{2} a^{2}-54 b a c +12 b^{3}+81 c^{2}}\right )^{{1}/{3}}}}-c_2 \,{\mathrm e}^{-\frac {\left (-\frac {i \sqrt {3}\, \textit {\_a}}{4}+\frac {3 \textit {\_a}}{4}-\frac {t}{2}\right ) \left (36 b a -108 c -8 a^{3}+12 \sqrt {12 c \,a^{3}-3 b^{2} a^{2}-54 b a c +12 b^{3}+81 c^{2}}\right )^{{2}/{3}}+\left (i \sqrt {3}\, \textit {\_a} +3 \textit {\_a} -2 t \right ) \left (a^{2}-3 b \right )}{3 \left (36 b a -108 c -8 a^{3}+12 \sqrt {12 c \,a^{3}-3 b^{2} a^{2}-54 b a c +12 b^{3}+81 c^{2}}\right )^{{1}/{3}}}}+c_1 \,{\mathrm e}^{\frac {\left (\frac {\left (-i \sqrt {3}-1\right ) \left (36 b a -108 c -8 a^{3}+12 \sqrt {12 c \,a^{3}-3 b^{2} a^{2}-54 b a c +12 b^{3}+81 c^{2}}\right )^{{2}/{3}}}{4}+\left (i \sqrt {3}-1\right ) \left (a^{2}-3 b \right )\right ) t}{3 \left (36 b a -108 c -8 a^{3}+12 \sqrt {12 c \,a^{3}-3 b^{2} a^{2}-54 b a c +12 b^{3}+81 c^{2}}\right )^{{1}/{3}}}}+c_2 \,{\mathrm e}^{\frac {\left (\frac {\left (i \sqrt {3}-1\right ) \left (36 b a -108 c -8 a^{3}+12 \sqrt {12 c \,a^{3}-3 b^{2} a^{2}-54 b a c +12 b^{3}+81 c^{2}}\right )^{{2}/{3}}}{4}+\left (-i \sqrt {3}-1\right ) \left (a^{2}-3 b \right )\right ) t}{3 \left (36 b a -108 c -8 a^{3}+12 \sqrt {12 c \,a^{3}-3 b^{2} a^{2}-54 b a c +12 b^{3}+81 c^{2}}\right )^{{1}/{3}}}}\right )
\]
✗ Mathematica
ode=D[x[t],{t,3}]+a*D[x[t],{t,2}]+b*D[x[t],t]+c*x[t]==0;
ic={x[Infinity]==0};
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
{}
✗ Sympy
from sympy import *
t = symbols("t")
a = symbols("a")
b = symbols("b")
c = symbols("c")
x = Function("x")
ode = Eq(a*Derivative(x(t), (t, 2)) + b*Derivative(x(t), t) + c*x(t) + Derivative(x(t), (t, 3)),0)
ics = {x(oo): 0}
dsolve(ode,func=x(t),ics=ics)
False