9.4.36 problem problem 47
Internal
problem
ID
[1000]
Book
:
Differential
equations
and
linear
algebra,
4th
ed.,
Edwards
and
Penney
Section
:
Section
7.3,
The
eigenvalue
method
for
linear
systems.
Page
395
Problem
number
:
problem
47
Date
solved
:
Tuesday, March 04, 2025 at 12:07:09 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=23 x_{1} \left (t \right )-18 x_{2} \left (t \right )-16 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-8 x_{1} \left (t \right )+6 x_{2} \left (t \right )+7 x_{3} \left (t \right )+9 x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=34 x_{1} \left (t \right )-27 x_{2} \left (t \right )-26 x_{3} \left (t \right )-9 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=-26 x_{1} \left (t \right )+21 x_{2} \left (t \right )+25 x_{3} \left (t \right )+12 x_{4} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.038 (sec). Leaf size: 123
ode:=[diff(x__1(t),t) = 23*x__1(t)-18*x__2(t)-16*x__3(t), diff(x__2(t),t) = -8*x__1(t)+6*x__2(t)+7*x__3(t)+9*x__4(t), diff(x__3(t),t) = 34*x__1(t)-27*x__2(t)-26*x__3(t)-9*x__4(t), diff(x__4(t),t) = -26*x__1(t)+21*x__2(t)+25*x__3(t)+12*x__4(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{6 t}+c_2 \,{\mathrm e}^{-3 t}+c_3 \,{\mathrm e}^{3 t}+c_4 \,{\mathrm e}^{9 t} \\
x_{2} \left (t \right ) &= \frac {c_1 \,{\mathrm e}^{6 t}}{2}+c_2 \,{\mathrm e}^{-3 t}+2 c_3 \,{\mathrm e}^{3 t}-c_4 \,{\mathrm e}^{9 t} \\
x_{3} \left (t \right ) &= \frac {c_1 \,{\mathrm e}^{6 t}}{2}+\frac {c_2 \,{\mathrm e}^{-3 t}}{2}-c_3 \,{\mathrm e}^{3 t}+2 c_4 \,{\mathrm e}^{9 t} \\
x_{4} \left (t \right ) &= \frac {c_1 \,{\mathrm e}^{6 t}}{2}-\frac {c_2 \,{\mathrm e}^{-3 t}}{2}+c_3 \,{\mathrm e}^{3 t}-c_4 \,{\mathrm e}^{9 t} \\
\end{align*}
✓ Mathematica. Time used: 0.007 (sec). Leaf size: 469
ode={D[ x1[t],t]==23*x1[t]-18*x2[t]-16*x3[t]+0*x4[t],D[ x2[t],t]==-8*x1[t]+6*x2[t]+7*x3[t]+9*x4[t],D[ x3[t],t]==34*x1[t]-27*x2[t]-26*x3[t]-9*x4[t],D[ x4[t],t]==-26*x1[t]+21*x2[t]+25*x3[t]+12*x4[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {x1}(t)\to \frac {1}{3} e^{-3 t} \left (c_1 \left (9 e^{6 t}-8 e^{9 t}+8 e^{12 t}-6\right )-\left (e^{3 t}-1\right ) \left (6 c_2 \left (e^{3 t}+e^{9 t}+1\right )+c_3 \left (6 e^{3 t}-3 e^{6 t}+7 e^{9 t}+6\right )+3 c_4 e^{6 t} \left (e^{3 t}-1\right )\right )\right ) \\
\text {x2}(t)\to \frac {1}{3} e^{-3 t} \left (-2 c_1 \left (-9 e^{6 t}+2 e^{9 t}+4 e^{12 t}+3\right )+3 c_2 \left (-4 e^{6 t}+e^{9 t}+2 e^{12 t}+2\right )+\left (e^{3 t}-1\right ) \left (c_3 \left (-6 e^{3 t}+12 e^{6 t}+7 e^{9 t}-6\right )+3 c_4 e^{6 t} \left (e^{3 t}+2\right )\right )\right ) \\
\text {x3}(t)\to \frac {1}{3} e^{-3 t} \left (c_1 \left (-9 e^{6 t}-4 e^{9 t}+16 e^{12 t}-3\right )+3 c_2 \left (2 e^{6 t}+e^{9 t}-4 e^{12 t}+1\right )+9 c_3 e^{6 t}+5 c_3 e^{9 t}-14 c_3 e^{12 t}+3 c_4 e^{6 t}+3 c_4 e^{9 t}-6 c_4 e^{12 t}+3 c_3\right ) \\
\text {x4}(t)\to \frac {1}{3} e^{-3 t} \left (c_1 \left (9 e^{6 t}-4 e^{9 t}-8 e^{12 t}+3\right )+3 c_2 \left (-2 e^{6 t}+e^{9 t}+2 e^{12 t}-1\right )-9 c_3 e^{6 t}+5 c_3 e^{9 t}+7 c_3 e^{12 t}-3 c_4 e^{6 t}+3 c_4 e^{9 t}+3 c_4 e^{12 t}-3 c_3\right ) \\
\end{align*}
✓ Sympy. Time used: 0.253 (sec). Leaf size: 124
from sympy import *
t = symbols("t")
x__1 = Function("x__1")
x__2 = Function("x__2")
x__3 = Function("x__3")
x__4 = Function("x__4")
ode=[Eq(-23*x__1(t) + 18*x__2(t) + 16*x__3(t) + Derivative(x__1(t), t),0),Eq(8*x__1(t) - 6*x__2(t) - 7*x__3(t) - 9*x__4(t) + Derivative(x__2(t), t),0),Eq(-34*x__1(t) + 27*x__2(t) + 26*x__3(t) + 9*x__4(t) + Derivative(x__3(t), t),0),Eq(26*x__1(t) - 21*x__2(t) - 25*x__3(t) - 12*x__4(t) + Derivative(x__4(t), t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t),x__3(t),x__4(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = - 2 C_{1} e^{- 3 t} + C_{2} e^{3 t} + 2 C_{3} e^{6 t} - C_{4} e^{9 t}, \ x^{2}{\left (t \right )} = - 2 C_{1} e^{- 3 t} + 2 C_{2} e^{3 t} + C_{3} e^{6 t} + C_{4} e^{9 t}, \ x^{3}{\left (t \right )} = - C_{1} e^{- 3 t} - C_{2} e^{3 t} + C_{3} e^{6 t} - 2 C_{4} e^{9 t}, \ x^{4}{\left (t \right )} = C_{1} e^{- 3 t} + C_{2} e^{3 t} + C_{3} e^{6 t} + C_{4} e^{9 t}\right ]
\]