9.6.29 problem problem 29
Internal
problem
ID
[1036]
Book
:
Differential
equations
and
linear
algebra,
4th
ed.,
Edwards
and
Penney
Section
:
Section
7.6,
Multiple
Eigenvalue
Solutions.
Page
451
Problem
number
:
problem
29
Date
solved
:
Saturday, March 29, 2025 at 10:37:23 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-x_{1} \left (t \right )+x_{2} \left (t \right )+x_{3} \left (t \right )-2 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=7 x_{1} \left (t \right )-4 x_{2} \left (t \right )-6 x_{3} \left (t \right )+11 x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=5 x_{1} \left (t \right )-x_{2} \left (t \right )+x_{3} \left (t \right )+3 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=6 x_{1} \left (t \right )-2 x_{2} \left (t \right )-2 x_{3} \left (t \right )+6 x_{4} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.234 (sec). Leaf size: 119
ode:=[diff(x__1(t),t) = -x__1(t)+x__2(t)+x__3(t)-2*x__4(t), diff(x__2(t),t) = 7*x__1(t)-4*x__2(t)-6*x__3(t)+11*x__4(t), diff(x__3(t),t) = 5*x__1(t)-x__2(t)+x__3(t)+3*x__4(t), diff(x__4(t),t) = 6*x__1(t)-2*x__2(t)-2*x__3(t)+6*x__4(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= {\mathrm e}^{-t} \left (c_4 t +c_3 \right ) \\
x_{2} \left (t \right ) &= -3 c_4 \,{\mathrm e}^{-t} t -3 c_3 \,{\mathrm e}^{-t}+c_4 \,{\mathrm e}^{-t}+{\mathrm e}^{2 t} t c_1 +{\mathrm e}^{2 t} c_2 \\
x_{3} \left (t \right ) &= -c_4 \,{\mathrm e}^{-t} t -c_3 \,{\mathrm e}^{-t}-{\mathrm e}^{2 t} t c_1 -2 \,{\mathrm e}^{2 t} c_1 -{\mathrm e}^{2 t} c_2 \\
x_{4} \left (t \right ) &= -2 c_4 \,{\mathrm e}^{-t} t -2 c_3 \,{\mathrm e}^{-t}-{\mathrm e}^{2 t} c_1 \\
\end{align*}
✓ Mathematica. Time used: 0.01 (sec). Leaf size: 196
ode={D[ x1[t],t]==-1*x1[t]+1*x2[t]+1*x3[t]-2*x4[t],D[ x2[t],t]==7*x1[t]-4*x2[t]-6*x3[t]+11*x4[t],D[ x3[t],t]==5*x1[t]-1*x2[t]+1*x3[t]+3*x4[t],D[ x4[t],t]==6*x1[t]-2*x2[t]-2*x3[t]+6*x4[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {x1}(t)\to e^{-t} ((c_2+c_3-2 c_4) t+c_1) \\
\text {x2}(t)\to e^{-t} \left (c_1 \left (e^{3 t} (3-2 t)-3\right )-3 c_2 t-c_3 e^{3 t}-3 c_3 t+2 c_4 e^{3 t}-c_4 e^{3 t} t+6 c_4 t+c_2+c_3-2 c_4\right ) \\
\text {x3}(t)\to e^{-t} \left (c_1 \left (e^{3 t} (2 t+1)-1\right )+c_3 e^{3 t}-t \left (-c_4 \left (e^{3 t}+2\right )+c_2+c_3\right )\right ) \\
\text {x4}(t)\to e^{-t} \left (2 c_1 \left (e^{3 t}-1\right )-2 (c_2+c_3-2 c_4) t+c_4 e^{3 t}\right ) \\
\end{align*}
✓ Sympy. Time used: 0.222 (sec). Leaf size: 104
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(x__1(t) - x__2(t) - x__3(t) + 2*x__4(t) + Derivative(x__1(t), t),0),Eq(-7*x__1(t) + 4*x__2(t) + 6*x__3(t) - 11*x__4(t) + Derivative(x__2(t), t),0),Eq(-5*x__1(t) + x__2(t) - x__3(t) - 3*x__4(t) + Derivative(x__3(t), t),0),Eq(-6*x__1(t) + 2*x__2(t) + 2*x__3(t) - 6*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)
\[
\left [ x^{1}{\left (t \right )} = C_{1} e^{- t} + C_{2} t e^{- t}, \ x^{2}{\left (t \right )} = - 3 C_{2} t e^{- t} - C_{3} t e^{2 t} - \left (3 C_{1} - C_{2}\right ) e^{- t} + \left (2 C_{3} - C_{4}\right ) e^{2 t}, \ x^{3}{\left (t \right )} = - C_{1} e^{- t} - C_{2} t e^{- t} + C_{3} t e^{2 t} + C_{4} e^{2 t}, \ x^{4}{\left (t \right )} = - 2 C_{1} e^{- t} - 2 C_{2} t e^{- t} + C_{3} e^{2 t}\right ]
\]