80.11.13 problem 13
Internal
problem
ID
[21421]
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
:
13
Date
solved
:
Thursday, October 02, 2025 at 07:31:05 PM
CAS
classification
:
system_of_ODEs
\begin{align*} x_{1}^{\prime }\left (t \right )&=x_{2} \left (t \right )+x_{4} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-x_{2} \left (t \right )+x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=x_{2} \left (t \right )+x_{3} \left (t \right )\\ x_{4}^{\prime }\left (t \right )&=x_{1} \left (t \right )-x_{4} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.159 (sec). Leaf size: 766
ode:=[diff(x__1(t),t) = x__2(t)+x__4(t), diff(x__2(t),t) = -x__2(t)+x__3(t), diff(x__3(t),t) = x__2(t)+x__3(t), diff(x__4(t),t) = x__1(t)-x__4(t)];
dsolve(ode);
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Mathematica. Time used: 0.024 (sec). Leaf size: 566
ode={D[x1[t],t]==x2[t]+x4[t],D[x2[t],t]==-x2[t]+x3[t],D[x3[t],t]==x2[t]+x3[t],D[x4[t],t]==x1[t]-x4[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
\begin{align*} \text {x1}(t)&\to \frac {1}{20} e^{-\frac {1}{2} \left (1+2 \sqrt {2}+\sqrt {5}\right ) t} \left (5 \left (\left (2+\sqrt {2}\right ) c_2-\sqrt {2} c_3\right ) e^{\frac {1}{2} \left (1+\sqrt {5}\right ) t}+\left (5 \sqrt {2} c_3-5 \left (\sqrt {2}-2\right ) c_2\right ) e^{\frac {1}{2} \left (1+4 \sqrt {2}+\sqrt {5}\right ) t}+2 \left (\left (5+\sqrt {5}\right ) c_1+\left (3 \sqrt {5}-5\right ) c_2+2 \sqrt {5} (c_4-c_3)\right ) e^{\left (\sqrt {2}+\sqrt {5}\right ) t}-2 \left (\left (\sqrt {5}-5\right ) c_1+\left (5+3 \sqrt {5}\right ) c_2+2 \sqrt {5} (c_4-c_3)\right ) e^{\sqrt {2} t}\right )\\ \text {x2}(t)&\to \frac {1}{4} e^{-\sqrt {2} t} \left (c_2 \left (-\left (\sqrt {2}-2\right ) e^{2 \sqrt {2} t}+2+\sqrt {2}\right )+\sqrt {2} c_3 \left (e^{2 \sqrt {2} t}-1\right )\right )\\ \text {x3}(t)&\to \frac {1}{4} e^{-\sqrt {2} t} \left (\sqrt {2} c_2 \left (e^{2 \sqrt {2} t}-1\right )+c_3 \left (\left (2+\sqrt {2}\right ) e^{2 \sqrt {2} t}+2-\sqrt {2}\right )\right )\\ \text {x4}(t)&\to \frac {1}{20} e^{-\frac {1}{2} \left (1+2 \sqrt {2}+\sqrt {5}\right ) t} \left (5 \left (\left (3 \sqrt {2}-4\right ) c_2-\left (\sqrt {2}-2\right ) c_3\right ) e^{\frac {1}{2} \left (1+4 \sqrt {2}+\sqrt {5}\right ) t}-5 \left (\left (4+3 \sqrt {2}\right ) c_2-\left (2+\sqrt {2}\right ) c_3\right ) e^{\frac {1}{2} \left (1+\sqrt {5}\right ) t}+2 \left (2 \sqrt {5} c_1+\left (10-4 \sqrt {5}\right ) c_2+\left (\sqrt {5}-5\right ) (c_3-c_4)\right ) e^{\left (\sqrt {2}+\sqrt {5}\right ) t}-2 \left (2 \sqrt {5} c_1-2 \left (5+2 \sqrt {5}\right ) c_2+\left (5+\sqrt {5}\right ) (c_3-c_4)\right ) e^{\sqrt {2} t}\right ) \end{align*}
✓ Sympy. Time used: 0.282 (sec). Leaf size: 204
from sympy import *
t = symbols("t")
x1 = Function("x1")
x2 = Function("x2")
x3 = Function("x3")
x4 = Function("x4")
ode=[Eq(-x2(t) - x4(t) + Derivative(x1(t), t),0),Eq(x2(t) - x3(t) + Derivative(x2(t), t),0),Eq(-x2(t) - x3(t) + Derivative(x3(t), t),0),Eq(-x1(t) + x4(t) + Derivative(x4(t), t),0)]
ics = {}
dsolve(ode,func=[x1(t),x2(t),x3(t),x4(t)],ics=ics)
\[
\left [ x_{1}{\left (t \right )} = \frac {C_{1} \left (1 - \sqrt {5}\right ) e^{- \frac {t \left (1 + \sqrt {5}\right )}{2}}}{2} + \frac {C_{2} \left (1 + \sqrt {5}\right ) e^{- \frac {t \left (1 - \sqrt {5}\right )}{2}}}{2} + C_{3} \left (1 - \sqrt {2}\right ) e^{- \sqrt {2} t} + C_{4} \left (1 + \sqrt {2}\right ) e^{\sqrt {2} t}, \ x_{2}{\left (t \right )} = C_{3} \left (1 - \sqrt {2}\right ) e^{- \sqrt {2} t} + C_{4} \left (1 + \sqrt {2}\right ) e^{\sqrt {2} t}, \ x_{3}{\left (t \right )} = C_{3} \left (3 - 2 \sqrt {2}\right ) e^{- \sqrt {2} t} + C_{4} \left (2 \sqrt {2} + 3\right ) e^{\sqrt {2} t}, \ x_{4}{\left (t \right )} = C_{1} e^{- \frac {t \left (1 + \sqrt {5}\right )}{2}} + C_{2} e^{- \frac {t \left (1 - \sqrt {5}\right )}{2}} + C_{3} e^{- \sqrt {2} t} + C_{4} e^{\sqrt {2} t}\right ]
\]