7.25.25 problem 25
Internal
problem
ID
[645]
Book
:
Elementary
Differential
Equations.
By
C.
Henry
Edwards,
David
E.
Penney
and
David
Calvis.
6th
edition.
2008
Section
:
Chapter
5.
Linear
systems
of
differential
equations.
Section
5.4
(The
eigenvalue
method
for
homogeneous
systems).
Problems
at
page
378
Problem
number
:
25
Date
solved
:
Saturday, March 29, 2025 at 05:01:27 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=5 x_{1} \left (t \right )+5 x_{2} \left (t \right )+2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-6 x_{1} \left (t \right )-6 x_{2} \left (t \right )-5 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=6 x_{1} \left (t \right )+6 x_{2} \left (t \right )+5 x_{3} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.137 (sec). Leaf size: 110
ode:=[diff(x__1(t),t) = 5*x__1(t)+5*x__2(t)+2*x__3(t), diff(x__2(t),t) = -6*x__1(t)-6*x__2(t)-5*x__3(t), diff(x__3(t),t) = 6*x__1(t)+6*x__2(t)+5*x__3(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= c_1 +c_2 \,{\mathrm e}^{2 t} \sin \left (3 t \right )+c_3 \,{\mathrm e}^{2 t} \cos \left (3 t \right ) \\
x_{2} \left (t \right ) &= -c_2 \,{\mathrm e}^{2 t} \sin \left (3 t \right )+c_2 \,{\mathrm e}^{2 t} \cos \left (3 t \right )-c_3 \,{\mathrm e}^{2 t} \cos \left (3 t \right )-c_3 \,{\mathrm e}^{2 t} \sin \left (3 t \right )-c_1 \\
x_{3} \left (t \right ) &= {\mathrm e}^{2 t} \left (\sin \left (3 t \right ) c_2 +\sin \left (3 t \right ) c_3 -\cos \left (3 t \right ) c_2 +\cos \left (3 t \right ) c_3 \right ) \\
\end{align*}
✓ Mathematica. Time used: 0.008 (sec). Leaf size: 122
ode={D[x1[t],t]==5*x1[t]+5*x2[t]+2*x3[t],D[x2[t],t]==-6*x1[t]-6*x2[t]-5*x3[t],D[x3[t],t]==6*x1[t]+6*x2[t]+5*x3[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {x1}(t)\to (c_1+c_2+c_3) e^{2 t} \cos (3 t)+(c_1+c_2) e^{2 t} \sin (3 t)-c_2-c_3 \\
\text {x2}(t)\to -c_3 e^{2 t} \cos (3 t)-(2 c_1+2 c_2+c_3) e^{2 t} \sin (3 t)+c_2+c_3 \\
\text {x3}(t)\to e^{2 t} (c_3 \cos (3 t)+(2 c_1+2 c_2+c_3) \sin (3 t)) \\
\end{align*}
✓ Sympy. Time used: 0.171 (sec). Leaf size: 90
from sympy import *
t = symbols("t")
x__1 = Function("x__1")
x__2 = Function("x__2")
x__3 = Function("x__3")
ode=[Eq(-5*x__1(t) - 5*x__2(t) - 2*x__3(t) + Derivative(x__1(t), t),0),Eq(6*x__1(t) + 6*x__2(t) + 5*x__3(t) + Derivative(x__2(t), t),0),Eq(-6*x__1(t) - 6*x__2(t) - 5*x__3(t) + Derivative(x__3(t), t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = - C_{1} + \left (\frac {C_{2}}{2} - \frac {C_{3}}{2}\right ) e^{2 t} \cos {\left (3 t \right )} - \left (\frac {C_{2}}{2} + \frac {C_{3}}{2}\right ) e^{2 t} \sin {\left (3 t \right )}, \ x^{2}{\left (t \right )} = C_{1} - C_{2} e^{2 t} \cos {\left (3 t \right )} + C_{3} e^{2 t} \sin {\left (3 t \right )}, \ x^{3}{\left (t \right )} = C_{2} e^{2 t} \cos {\left (3 t \right )} - C_{3} e^{2 t} \sin {\left (3 t \right )}\right ]
\]