20.16.10 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
:
Monday, January 27, 2025 at 08:03:18 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*}
✓ Solution by Maple
Time used: 0.048 (sec). Leaf size: 60
dsolve([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)],singsol=all)
\begin{align*}
x_{1} \left (t \right ) &= c_{1} {\mathrm e}^{-3 t}+c_{2} {\mathrm e}^{t}+{\mathrm e}^{2 t} c_3 \\
x_{2} \left (t \right ) &= 2 c_{2} {\mathrm e}^{t}-5 \,{\mathrm e}^{2 t} c_3 \\
x_{3} \left (t \right ) &= -c_{1} {\mathrm e}^{-3 t}-c_{2} {\mathrm e}^{t}+\frac {3 \,{\mathrm e}^{2 t} c_3}{2} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.018 (sec). Leaf size: 460
DSolve[{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]},{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*}