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
:
Tuesday, March 04, 2025 at 05:19:11 PM
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*}
✓ Maple. Time used: 0.059 (sec). Leaf size: 136
ode:=[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)];
dsolve(ode);
\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*}
✓ Mathematica. Time used: 0.049 (sec). Leaf size: 259
ode={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]};
ic={};
DSolve[{ode,ic},{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*}
✓ Sympy. Time used: 0.299 (sec). Leaf size: 143
from sympy import *
t = symbols("t")
x__1 = Function("x__1")
x__2 = Function("x__2")
x__3 = Function("x__3")
ode=[Eq(16*x__1(t) - 30*x__2(t) + 18*x__3(t) + Derivative(x__1(t), t),0),Eq(8*x__1(t) - 8*x__2(t) - 16*x__3(t) + Derivative(x__2(t), t),0),Eq(-8*x__1(t) + 15*x__2(t) - 9*x__3(t) + Derivative(x__3(t), t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = \frac {39 C_{1}}{7} + 2 C_{2} e^{\frac {t}{2}} \sin {\left (\frac {\sqrt {1695} t}{2} \right )} - 2 C_{3} e^{\frac {t}{2}} \cos {\left (\frac {\sqrt {1695} t}{2} \right )}, \ x^{2}{\left (t \right )} = \frac {25 C_{1}}{7} + \left (\frac {C_{2}}{2} + \frac {\sqrt {1695} C_{3}}{30}\right ) e^{\frac {t}{2}} \sin {\left (\frac {\sqrt {1695} t}{2} \right )} + \left (\frac {\sqrt {1695} C_{2}}{30} - \frac {C_{3}}{2}\right ) e^{\frac {t}{2}} \cos {\left (\frac {\sqrt {1695} t}{2} \right )}, \ x^{3}{\left (t \right )} = C_{1} - C_{2} e^{\frac {t}{2}} \sin {\left (\frac {\sqrt {1695} t}{2} \right )} + C_{3} e^{\frac {t}{2}} \cos {\left (\frac {\sqrt {1695} t}{2} \right )}\right ]
\]