14.16.10 problem 10
Internal
problem
ID
[3843]
Book
:
Differential
equations
and
linear
algebra,
Stephen
W.
Goode
and
Scott
A
Annin.
Fourth
edition,
2015
Section
:
Chapter
9,
First
order
linear
systems.
Section
9.4
(Nondefective
coefficient
matrix),
page
607
Problem
number
:
10
Date
solved
:
Tuesday, September 30, 2025 at 06:57:59 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=3 x_{1} \left (t \right )+2 x_{2} \left (t \right )+6 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-2 x_{1} \left (t \right )+x_{2} \left (t \right )-2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-x_{1} \left (t \right )-2 x_{2} \left (t \right )-4 x_{3} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.115 (sec). Leaf size: 60
ode:=[diff(x__1(t),t) = 3*x__1(t)+2*x__2(t)+6*x__3(t), diff(x__2(t),t) = -2*x__1(t)+x__2(t)-2*x__3(t), diff(x__3(t),t) = -x__1(t)-2*x__2(t)-4*x__3(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{t}+c_2 \,{\mathrm e}^{-3 t}+c_3 \,{\mathrm e}^{2 t} \\
x_{2} \left (t \right ) &= 2 c_1 \,{\mathrm e}^{t}-5 c_3 \,{\mathrm e}^{2 t} \\
x_{3} \left (t \right ) &= -c_1 \,{\mathrm e}^{t}-c_2 \,{\mathrm e}^{-3 t}+\frac {3 c_3 \,{\mathrm e}^{2 t}}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.011 (sec). Leaf size: 460
ode={D[x1[t],t]==3*x1[t]+2*x2[t]+6*x3[t],D[x2[t],t]==-2*x1[t]+x2[t]-2*x3[t],D[x3[t],t]==x1[t]-2*x2[t]-4*x3[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
\begin{align*} \text {x1}(t)&\to 2 c_2 \text {RootSum}\left [\text {$\#$1}^3-19 \text {$\#$1}+26\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-2 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-19}\&\right ]+2 c_3 \text {RootSum}\left [\text {$\#$1}^3-19 \text {$\#$1}+26\&,\frac {3 \text {$\#$1} e^{\text {$\#$1} t}-5 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-19}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3-19 \text {$\#$1}+26\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+3 \text {$\#$1} e^{\text {$\#$1} t}-8 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-19}\&\right ]\\ \text {x2}(t)&\to -2 c_3 \text {RootSum}\left [\text {$\#$1}^3-19 \text {$\#$1}+26\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}+3 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-19}\&\right ]-2 c_1 \text {RootSum}\left [\text {$\#$1}^3-19 \text {$\#$1}+26\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}+5 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-19}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3-19 \text {$\#$1}+26\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}+\text {$\#$1} e^{\text {$\#$1} t}-18 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-19}\&\right ]\\ \text {x3}(t)&\to -2 c_2 \text {RootSum}\left [\text {$\#$1}^3-19 \text {$\#$1}+26\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-4 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-19}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3-19 \text {$\#$1}+26\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}+3 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-19}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3-19 \text {$\#$1}+26\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-4 \text {$\#$1} e^{\text {$\#$1} t}+7 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-19}\&\right ] \end{align*}
✓ Sympy. Time used: 0.092 (sec). Leaf size: 65
from sympy import *
t = symbols("t")
x__1 = Function("x__1")
x__2 = Function("x__2")
x__3 = Function("x__3")
ode=[Eq(-3*x__1(t) - 2*x__2(t) - 6*x__3(t) + Derivative(x__1(t), t),0),Eq(2*x__1(t) - x__2(t) + 2*x__3(t) + Derivative(x__2(t), t),0),Eq(x__1(t) + 2*x__2(t) + 4*x__3(t) + Derivative(x__3(t), t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = - C_{1} e^{- 3 t} - C_{2} e^{t} + \frac {2 C_{3} e^{2 t}}{3}, \ x^{2}{\left (t \right )} = - 2 C_{2} e^{t} - \frac {10 C_{3} e^{2 t}}{3}, \ x^{3}{\left (t \right )} = C_{1} e^{- 3 t} + C_{2} e^{t} + C_{3} e^{2 t}\right ]
\]