80.7.28 problem C 6
Internal
problem
ID
[21347]
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
7.
System
of
first
order
equations.
Excercise
7.6
at
page
162
Problem
number
:
C
6
Date
solved
:
Thursday, October 02, 2025 at 07:28:40 PM
CAS
classification
:
[[_3rd_order, _missing_x]]
\begin{align*} x^{\prime \prime \prime }-2 x^{\prime \prime }+3 x^{\prime }+x&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 149
ode:=diff(diff(diff(x(t),t),t),t)-2*diff(diff(x(t),t),t)+3*diff(x(t),t)+x(t) = 0;
dsolve(ode,x(t), singsol=all);
\[
x = {\mathrm e}^{\frac {\left (\left (260+60 \sqrt {21}\right )^{{1}/{3}}+10\right ) \left (\left (260+60 \sqrt {21}\right )^{{1}/{3}}-2\right ) t}{12 \left (260+60 \sqrt {21}\right )^{{1}/{3}}}} \left (\sin \left (\frac {\sqrt {3}\, \left (\left (260+60 \sqrt {3}\, \sqrt {7}\right )^{{2}/{3}}+20\right ) t}{12 \left (260+60 \sqrt {3}\, \sqrt {7}\right )^{{1}/{3}}}\right ) c_2 +\cos \left (\frac {\sqrt {3}\, \left (\left (260+60 \sqrt {3}\, \sqrt {7}\right )^{{2}/{3}}+20\right ) t}{12 \left (260+60 \sqrt {3}\, \sqrt {7}\right )^{{1}/{3}}}\right ) c_3 \right )+c_1 \,{\mathrm e}^{-\frac {\left (\left (260+60 \sqrt {21}\right )^{{2}/{3}}-4 \left (260+60 \sqrt {21}\right )^{{1}/{3}}-20\right ) t}{6 \left (260+60 \sqrt {21}\right )^{{1}/{3}}}}
\]
✓ Mathematica. Time used: 0.005 (sec). Leaf size: 87
ode=D[x[t],{t,3}]-2*D[x[t],{t,2}]+3*D[x[t],t]+x[t]==0;
ic={};
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to c_1 \exp \left (t \text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}+1\&,1\right ]\right )+c_2 \exp \left (t \text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}+1\&,2\right ]\right )+c_3 \exp \left (t \text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+3 \text {$\#$1}+1\&,3\right ]\right ) \end{align*}
✓ Sympy. Time used: 0.383 (sec). Leaf size: 267
from sympy import *
t = symbols("t")
x = Function("x")
ode = Eq(x(t) + 3*Derivative(x(t), t) - 2*Derivative(x(t), (t, 2)) + Derivative(x(t), (t, 3)),0)
ics = {}
dsolve(ode,func=x(t),ics=ics)
\[
x{\left (t \right )} = C_{1} e^{\frac {t \left (- \frac {2 \sqrt [3]{2} \cdot 5^{\frac {2}{3}}}{\sqrt [3]{13 + 3 \sqrt {21}}} + 8 + 2^{\frac {2}{3}} \sqrt [3]{5} \sqrt [3]{13 + 3 \sqrt {21}}\right )}{12}} \sin {\left (\frac {\sqrt [3]{10} \sqrt {3} t \left (\frac {2 \sqrt [3]{5}}{\sqrt [3]{13 + 3 \sqrt {21}}} + \sqrt [3]{2} \sqrt [3]{13 + 3 \sqrt {21}}\right )}{12} \right )} + C_{2} e^{\frac {t \left (- \frac {2 \sqrt [3]{2} \cdot 5^{\frac {2}{3}}}{\sqrt [3]{13 + 3 \sqrt {21}}} + 8 + 2^{\frac {2}{3}} \sqrt [3]{5} \sqrt [3]{13 + 3 \sqrt {21}}\right )}{12}} \cos {\left (\frac {\sqrt [3]{10} \sqrt {3} t \left (\frac {2 \sqrt [3]{5}}{\sqrt [3]{13 + 3 \sqrt {21}}} + \sqrt [3]{2} \sqrt [3]{13 + 3 \sqrt {21}}\right )}{12} \right )} + C_{3} e^{\frac {t \left (- 2^{\frac {2}{3}} \sqrt [3]{5} \sqrt [3]{13 + 3 \sqrt {21}} + \frac {2 \sqrt [3]{2} \cdot 5^{\frac {2}{3}}}{\sqrt [3]{13 + 3 \sqrt {21}}} + 4\right )}{6}}
\]