9.4.35 problem problem 46
Internal
problem
ID
[999]
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
46
Date
solved
:
Tuesday, March 04, 2025 at 12:07:08 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=13 x_{1} \left (t \right )-42 x_{2} \left (t \right )+106 x_{3} \left (t \right )+139 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )-16 x_{2} \left (t \right )+52 x_{3} \left (t \right )+70 x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{1} \left (t \right )+6 x_{2} \left (t \right )-20 x_{3} \left (t \right )-31 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=-x_{1} \left (t \right )-6 x_{2} \left (t \right )+22 x_{3} \left (t \right )+33 x_{4} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.374 (sec). Leaf size: 122
ode:=[diff(x__1(t),t) = 13*x__1(t)-42*x__2(t)+106*x__3(t)+139*x__4(t), diff(x__2(t),t) = 2*x__1(t)-16*x__2(t)+52*x__3(t)+70*x__4(t), diff(x__3(t),t) = x__1(t)+6*x__2(t)-20*x__3(t)-31*x__4(t), diff(x__4(t),t) = -x__1(t)-6*x__2(t)+22*x__3(t)+33*x__4(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= {\mathrm e}^{2 t} c_1 +c_2 \,{\mathrm e}^{-4 t}+c_3 \,{\mathrm e}^{8 t}+c_4 \,{\mathrm e}^{4 t} \\
x_{2} \left (t \right ) &= 2 \,{\mathrm e}^{2 t} c_1 +\frac {2 c_2 \,{\mathrm e}^{-4 t}}{3}-\frac {2 c_3 \,{\mathrm e}^{8 t}}{3}+c_4 \,{\mathrm e}^{4 t} \\
x_{3} \left (t \right ) &= 2 \,{\mathrm e}^{2 t} c_1 -\frac {c_2 \,{\mathrm e}^{-4 t}}{3}+c_3 \,{\mathrm e}^{8 t}-c_4 \,{\mathrm e}^{4 t} \\
x_{4} \left (t \right ) &= -{\mathrm e}^{2 t} c_1 +\frac {c_2 \,{\mathrm e}^{-4 t}}{3}-c_3 \,{\mathrm e}^{8 t}+c_4 \,{\mathrm e}^{4 t} \\
\end{align*}
✓ Mathematica. Time used: 0.007 (sec). Leaf size: 449
ode={D[ x1[t],t]==13*x1[t]-42*x2[t]+106*x3[t]+139*x4[t],D[ x2[t],t]==2*x1[t]-16*x2[t]+52*x3[t]+70*x4[t],D[ x3[t],t]==1*x1[t]+6*x2[t]-20*x3[t]-31*x4[t],D[ x4[t],t]==-1*x1[t]-6*x2[t]+22*x3[t]+33*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}{4} e^{-4 t} \left (c_1 \left (4 e^{8 t}+3 e^{12 t}-3\right )-6 c_2 \left (2 e^{8 t}+e^{12 t}-3\right )+4 c_3 e^{6 t}+32 c_3 e^{8 t}+12 c_3 e^{12 t}+4 c_4 e^{6 t}+44 c_4 e^{8 t}+15 c_4 e^{12 t}-48 c_3-63 c_4\right ) \\
\text {x2}(t)\to \frac {1}{2} e^{-4 t} \left (-\left (c_1 \left (-2 e^{8 t}+e^{12 t}+1\right )\right )+2 c_2 \left (-3 e^{8 t}+e^{12 t}+3\right )+4 c_3 e^{6 t}+16 c_3 e^{8 t}-4 c_3 e^{12 t}+4 c_4 e^{6 t}+22 c_4 e^{8 t}-5 c_4 e^{12 t}-16 c_3-21 c_4\right ) \\
\text {x3}(t)\to \frac {1}{4} e^{-4 t} \left (c_1 \left (-4 e^{8 t}+3 e^{12 t}+1\right )-6 c_2 \left (-2 e^{8 t}+e^{12 t}+1\right )+8 c_3 e^{6 t}-32 c_3 e^{8 t}+12 c_3 e^{12 t}+8 c_4 e^{6 t}-44 c_4 e^{8 t}+15 c_4 e^{12 t}+16 c_3+21 c_4\right ) \\
\text {x4}(t)\to \frac {1}{4} e^{-4 t} \left (c_1 \left (4 e^{8 t}-3 e^{12 t}-1\right )+6 c_2 \left (-2 e^{8 t}+e^{12 t}+1\right )-4 c_3 e^{6 t}+32 c_3 e^{8 t}-12 c_3 e^{12 t}-4 c_4 e^{6 t}+44 c_4 e^{8 t}-15 c_4 e^{12 t}-16 c_3-21 c_4\right ) \\
\end{align*}
✓ Sympy. Time used: 0.240 (sec). Leaf size: 128
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(-13*x__1(t) + 42*x__2(t) - 106*x__3(t) - 139*x__4(t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) + 16*x__2(t) - 52*x__3(t) - 70*x__4(t) + Derivative(x__2(t), t),0),Eq(-x__1(t) - 6*x__2(t) + 20*x__3(t) + 31*x__4(t) + Derivative(x__3(t), t),0),Eq(x__1(t) + 6*x__2(t) - 22*x__3(t) - 33*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)
\[
\left [ x^{1}{\left (t \right )} = 3 C_{1} e^{- 4 t} - C_{2} e^{2 t} + C_{3} e^{4 t} - C_{4} e^{8 t}, \ x^{2}{\left (t \right )} = 2 C_{1} e^{- 4 t} - 2 C_{2} e^{2 t} + C_{3} e^{4 t} + \frac {2 C_{4} e^{8 t}}{3}, \ x^{3}{\left (t \right )} = - C_{1} e^{- 4 t} - 2 C_{2} e^{2 t} - C_{3} e^{4 t} - C_{4} e^{8 t}, \ x^{4}{\left (t \right )} = C_{1} e^{- 4 t} + C_{2} e^{2 t} + C_{3} e^{4 t} + C_{4} e^{8 t}\right ]
\]