76.29.8 problem 8
Internal
problem
ID
[17813]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
6.
Systems
of
First
Order
Linear
Equations.
Section
6.7
(Defective
Matrices).
Problems
at
page
444
Problem
number
:
8
Date
solved
:
Monday, March 31, 2025 at 04:33:11 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 )-2 x_{3} \left (t \right )+3 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )-\frac {3 x_{2} \left (t \right )}{2}-x_{3} \left (t \right )+\frac {7 x_{4} \left (t \right )}{2}\\ \frac {d}{d t}x_{3} \left (t \right )&=-x_{1} \left (t \right )+\frac {x_{2} \left (t \right )}{2}-\frac {3 x_{4} \left (t \right )}{2}\\ \frac {d}{d t}x_{4} \left (t \right )&=-2 x_{1} \left (t \right )+\frac {3 x_{2} \left (t \right )}{2}+3 x_{3} \left (t \right )-\frac {7 x_{4} \left (t \right )}{2} \end{align*}
✓ Maple. Time used: 0.190 (sec). Leaf size: 110
ode:=[diff(x__1(t),t) = x__1(t)-x__2(t)-2*x__3(t)+3*x__4(t), diff(x__2(t),t) = 2*x__1(t)-3/2*x__2(t)-x__3(t)+7/2*x__4(t), diff(x__3(t),t) = -x__1(t)+1/2*x__2(t)-3/2*x__4(t), diff(x__4(t),t) = -2*x__1(t)+3/2*x__2(t)+3*x__3(t)-7/2*x__4(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= {\mathrm e}^{-t} \left (-2 c_4 \,t^{2}-2 c_3 t +c_1 -2 c_2 \right ) \\
x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (-6 c_4 \,t^{2}-6 c_3 t -4 c_4 t +c_1 -6 c_2 -2 c_3 -6 c_4 \right )}{2} \\
x_{3} \left (t \right ) &= {\mathrm e}^{-t} \left (c_4 \,t^{2}+c_3 t +c_2 \right ) \\
x_{4} \left (t \right ) &= -\frac {{\mathrm e}^{-t} \left (-2 c_4 \,t^{2}-2 c_3 t +4 c_4 t +c_1 -2 c_2 +2 c_3 +2 c_4 \right )}{2} \\
\end{align*}
✓ Mathematica. Time used: 0.005 (sec). Leaf size: 206
ode={D[x1[t],t]==1*x1[t]-1*x2[t]-2*x3[t]+3*x4[t],D[x2[t],t]==2*x1[t]-3/2*x2[t]-1*x3[t]+7/2*x4[t],D[x3[t],t]==-1*x1[t]+1/2*x2[t]-0*x3[t]-3/2*x4[t],D[x4[t],t]==-2*x1[t]+3/2*x2[t]+3*x3[t]-7/2*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} \left (c_1 \left (-t^2+2 t+1\right )+t (c_2 (t-1)+2 c_3 (t-1)-c_4 (t-3))\right ) \\
\text {x2}(t)\to \frac {1}{2} e^{-t} \left (-3 (c_1-c_2-2 c_3+c_4) t^2+(4 c_1-c_2-2 c_3+7 c_4) t+2 c_2\right ) \\
\text {x3}(t)\to \frac {1}{2} e^{-t} \left ((c_1-c_2-2 c_3+c_4) t^2+(-2 c_1+c_2+2 c_3-3 c_4) t+2 c_3\right ) \\
\text {x4}(t)\to \frac {1}{2} e^{-t} \left ((c_1-c_2-2 c_3+c_4) t^2+(-4 c_1+3 c_2+6 c_3-5 c_4) t+2 c_4\right ) \\
\end{align*}
✓ Sympy. Time used: 0.243 (sec). Leaf size: 133
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) + 2*x__3(t) - 3*x__4(t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) + 3*x__2(t)/2 + x__3(t) - 7*x__4(t)/2 + Derivative(x__2(t), t),0),Eq(x__1(t) - x__2(t)/2 + 3*x__4(t)/2 + Derivative(x__3(t), t),0),Eq(2*x__1(t) - 3*x__2(t)/2 - 3*x__3(t) + 7*x__4(t)/2 + Derivative(x__4(t), t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t),x__3(t),x__4(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = - C_{2} t^{2} e^{- t} + t \left (2 C_{2} - 2 C_{3}\right ) e^{- t} + \left (- 2 C_{1} + C_{2} + 2 C_{3}\right ) e^{- t}, \ x^{2}{\left (t \right )} = - \frac {3 C_{2} t^{2} e^{- t}}{2} + t \left (2 C_{2} - 3 C_{3}\right ) e^{- t} - \left (3 C_{1} - 2 C_{3} + 2 C_{4}\right ) e^{- t}, \ x^{3}{\left (t \right )} = \frac {C_{2} t^{2} e^{- t}}{2} - t \left (C_{2} - C_{3}\right ) e^{- t} + \left (C_{1} - C_{3} + C_{4}\right ) e^{- t}, \ x^{4}{\left (t \right )} = \frac {C_{2} t^{2} e^{- t}}{2} - t \left (2 C_{2} - C_{3}\right ) e^{- t} + \left (C_{1} - 2 C_{3}\right ) e^{- t}\right ]
\]