20.19.7 problem 8
Internal
problem
ID
[3887]
Book
:
Differential
equations
and
linear
algebra,
Stephen
W.
Goode
and
Scott
A
Annin.
Fourth
edition,
2015
Section
:
Chapter
9,
First
order
linear
systems.
Section
9.8
(Matrix
exponential
function),
page
642
Problem
number
:
8
Date
solved
:
Monday, January 27, 2025 at 08:03:55 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{2} \left (t \right )+3 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )+3 x_{2} \left (t \right )-2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=2 x_{2} \left (t \right )+2 x_{3} \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 0.152 (sec). Leaf size: 1983
dsolve([diff(x__1(t),t)=0*x__1(t)+x__2(t)+3*x__3(t),diff(x__2(t),t)=2*x__1(t)+3*x__2(t)-2*x__3(t),diff(x__3(t),t)=1*x__2(t)+1*x__2(t)+2*x__3(t)],singsol=all)
\begin{align*}
\text {Expression too large to display} \\
x_{2} \left (t \right ) &= \left (\left (-\frac {\sqrt {3}\, \left (\left (53+6 \sqrt {78}\right )^{{2}/{3}}-1\right ) c_3}{12 \left (53+6 \sqrt {78}\right )^{{1}/{3}}}+\frac {\left (\left (53+6 \sqrt {78}\right )^{{2}/{3}}+2 \left (53+6 \sqrt {78}\right )^{{1}/{3}}+1\right ) c_{2}}{12 \left (53+6 \sqrt {78}\right )^{{1}/{3}}}\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (53+6 \sqrt {78}\right )^{{2}/{3}}-1\right ) t}{6 \left (53+6 \sqrt {78}\right )^{{1}/{3}}}\right )+\left (-\frac {\left (\left (53+6 \sqrt {78}\right )^{{2}/{3}}+2 \left (53+6 \sqrt {78}\right )^{{1}/{3}}+1\right ) c_3}{12 \left (53+6 \sqrt {78}\right )^{{1}/{3}}}-\frac {\sqrt {3}\, \left (\left (53+6 \sqrt {78}\right )^{{2}/{3}}-1\right ) c_{2}}{12 \left (53+6 \sqrt {78}\right )^{{1}/{3}}}\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (53+6 \sqrt {78}\right )^{{2}/{3}}-1\right ) t}{6 \left (53+6 \sqrt {78}\right )^{{1}/{3}}}\right )\right ) {\mathrm e}^{-\frac {\left (1+\left (53+6 \sqrt {78}\right )^{{2}/{3}}-10 \left (53+6 \sqrt {78}\right )^{{1}/{3}}\right ) t}{6 \left (53+6 \sqrt {78}\right )^{{1}/{3}}}}+\frac {\left (\left (53+6 \sqrt {78}\right )^{{2}/{3}}-\left (53+6 \sqrt {78}\right )^{{1}/{3}}+1\right ) c_{1} {\mathrm e}^{\frac {\left (\left (53+6 \sqrt {78}\right )^{{2}/{3}}+5 \left (53+6 \sqrt {78}\right )^{{1}/{3}}+1\right ) t}{3 \left (53+6 \sqrt {78}\right )^{{1}/{3}}}}}{6 \left (53+6 \sqrt {78}\right )^{{1}/{3}}} \\
x_{3} \left (t \right ) &= -c_{2} {\mathrm e}^{-\frac {\left (1+\left (53+6 \sqrt {78}\right )^{{2}/{3}}-10 \left (53+6 \sqrt {78}\right )^{{1}/{3}}\right ) t}{6 \left (53+6 \sqrt {78}\right )^{{1}/{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (53+6 \sqrt {78}\right )^{{2}/{3}}-1\right ) t}{6 \left (53+6 \sqrt {78}\right )^{{1}/{3}}}\right )+c_3 \,{\mathrm e}^{-\frac {\left (1+\left (53+6 \sqrt {78}\right )^{{2}/{3}}-10 \left (53+6 \sqrt {78}\right )^{{1}/{3}}\right ) t}{6 \left (53+6 \sqrt {78}\right )^{{1}/{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (53+6 \sqrt {78}\right )^{{2}/{3}}-1\right ) t}{6 \left (53+6 \sqrt {78}\right )^{{1}/{3}}}\right )+c_{1} {\mathrm e}^{\frac {\left (\left (53+6 \sqrt {78}\right )^{{2}/{3}}+5 \left (53+6 \sqrt {78}\right )^{{1}/{3}}+1\right ) t}{3 \left (53+6 \sqrt {78}\right )^{{1}/{3}}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 148
DSolve[{D[x1[t],t]==0*x1[t]+x2[t]+3*x3[t],D[x2[t],t]==2*x1[t]+3*x2[t]-2*x3[t],D[x3[t],t]==1*x1[t]+1*x2[t]+2*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
\text {x1}(t)\to \frac {1}{8} e^{-t} \left (c_1 \left (e^{4 t} (4 t+1)+7\right )+e^{4 t} (8 c_2 t-4 c_3 t+7 c_3)-7 c_3\right ) \\
\text {x2}(t)\to \frac {1}{2} e^{-t} \left (c_1 \left (e^{4 t}-1\right )+(2 c_2-c_3) e^{4 t}+c_3\right ) \\
\text {x3}(t)\to \frac {1}{8} e^{-t} \left (c_1 \left (e^{4 t} (4 t+1)-1\right )+e^{4 t} (8 c_2 t-4 c_3 t+7 c_3)+c_3\right ) \\
\end{align*}