20.20.16 problem 16
Internal
problem
ID
[3906]
Book
:
Differential
equations
and
linear
algebra,
Stephen
W.
Goode
and
Scott
A
Annin.
Fourth
edition,
2015
Section
:
Chapter
9,
First
order
linear
systems.
Section
9.11
(Chapter
review),
page
665
Problem
number
:
16
Date
solved
:
Monday, January 27, 2025 at 08:04:19 AM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-16 x_{1} \left (t \right )+30 x_{2} \left (t \right )-18 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-8 x_{1} \left (t \right )+8 x_{2} \left (t \right )+16 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=8 x_{1} \left (t \right )-15 x_{2} \left (t \right )+9 x_{3} \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 0.059 (sec). Leaf size: 136
dsolve([diff(x__1(t),t)=-16*x__1(t)+30*x__2(t)-18*x__3(t),diff(x__2(t),t)=-8*x__1(t)+8*x__2(t)+16*x__3(t),diff(x__3(t),t)=8*x__1(t)-15*x__2(t)+9*x__3(t)],singsol=all)
\begin{align*}
x_{1} \left (t \right ) &= c_{1} +c_{2} {\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {1695}\, t}{2}\right )+c_3 \,{\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {1695}\, t}{2}\right ) \\
x_{2} \left (t \right ) &= \frac {c_{2} {\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {1695}\, t}{2}\right )}{4}+\frac {c_{2} {\mathrm e}^{\frac {t}{2}} \sqrt {1695}\, \cos \left (\frac {\sqrt {1695}\, t}{2}\right )}{60}+\frac {c_3 \,{\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {1695}\, t}{2}\right )}{4}-\frac {c_3 \,{\mathrm e}^{\frac {t}{2}} \sqrt {1695}\, \sin \left (\frac {\sqrt {1695}\, t}{2}\right )}{60}+\frac {25 c_{1}}{39} \\
x_{3} \left (t \right ) &= -\frac {c_{2} {\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {1695}\, t}{2}\right )}{2}-\frac {c_3 \,{\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {1695}\, t}{2}\right )}{2}+\frac {7 c_{1}}{39} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.049 (sec). Leaf size: 259
DSolve[{D[x1[t],t]==-16*x1[t]+30*x2[t]-18*x3[t],D[x2[t],t]==-8*x1[t]+8*x2[t]+16*x3[t],D[x3[t],t]==8*x1[t]-15*x2[t]+9*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
\text {x1}(t)\to \frac {226 (7 c_1-39 c_3) e^{t/2} \cos \left (\frac {\sqrt {1695} t}{2}\right )-2 \sqrt {1695} (57 c_1-106 c_2+61 c_3) e^{t/2} \sin \left (\frac {\sqrt {1695} t}{2}\right )+4407 (c_1+2 c_3)}{5989} \\
\text {x2}(t)\to -\frac {1}{53} (25 c_1-53 c_2+50 c_3) e^{t/2} \cos \left (\frac {\sqrt {1695} t}{2}\right )-\frac {(823 c_1-795 c_2-1746 c_3) e^{t/2} \sin \left (\frac {\sqrt {1695} t}{2}\right )}{53 \sqrt {1695}}+\frac {25}{53} (c_1+2 c_3) \\
\text {x3}(t)\to \frac {-113 (7 c_1-39 c_3) e^{t/2} \cos \left (\frac {\sqrt {1695} t}{2}\right )+\sqrt {1695} (57 c_1-106 c_2+61 c_3) e^{t/2} \sin \left (\frac {\sqrt {1695} t}{2}\right )+791 (c_1+2 c_3)}{5989} \\
\end{align*}