20.17.12 problem 12
Internal
problem
ID
[3866]
Book
:
Differential
equations
and
linear
algebra,
Stephen
W.
Goode
and
Scott
A
Annin.
Fourth
edition,
2015
Section
:
Chapter
9,
First
order
linear
systems.
Section
9.5
(Defective
coefficient
matrix),
page
619
Problem
number
:
12
Date
solved
:
Tuesday, March 04, 2025 at 05:18:18 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{1} \left (t \right )+2 x_{3} \left (t \right )+x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=x_{2} \left (t \right )+2 x_{4} \left (t \right ) \end{align*}
✓ Maple. Time used: 0.260 (sec). Leaf size: 93
ode:=[diff(x__1(t),t) = -x__2(t), diff(x__2(t),t) = x__1(t), diff(x__3(t),t) = x__1(t)+2*x__3(t)+x__4(t), diff(x__4(t),t) = x__2(t)+2*x__4(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= \cos \left (t \right ) c_3 -c_4 \sin \left (t \right ) \\
x_{2} \left (t \right ) &= c_3 \sin \left (t \right )+c_4 \cos \left (t \right ) \\
x_{3} \left (t \right ) &= \frac {8 c_3 \sin \left (t \right )}{25}+\frac {6 c_4 \sin \left (t \right )}{25}-\frac {6 \cos \left (t \right ) c_3}{25}+\frac {8 c_4 \cos \left (t \right )}{25}+c_{2} {\mathrm e}^{2 t} t +{\mathrm e}^{2 t} c_{1} \\
x_{4} \left (t \right ) &= -\frac {\cos \left (t \right ) c_3}{5}-\frac {2 c_3 \sin \left (t \right )}{5}-\frac {2 c_4 \cos \left (t \right )}{5}+\frac {c_4 \sin \left (t \right )}{5}+c_{2} {\mathrm e}^{2 t} \\
\end{align*}
✓ Mathematica. Time used: 0.211 (sec). Leaf size: 260
ode={D[x1[t],t]==0*x1[t]-1*x2[t]+0*x3[t]+0*x4[t],D[x2[t],t]==1*x1[t]+0*x2[t]+0*x3[t]+0*x4[t],D[x3[t],t]==1*x1[t]+0*x2[t]+2*x3[t]+1*x4[t],D[x4[t],t]==0*x1[t]+1*x2[t]+0*x3[t]+2*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}{2} e^{-i t} \left (c_1 \left (1+e^{2 i t}\right )+i c_2 \left (-1+e^{2 i t}\right )\right ) \\
\text {x2}(t)\to \frac {1}{2} e^{-i t} \left (c_2 \left (1+e^{2 i t}\right )-i c_1 \left (-1+e^{2 i t}\right )\right ) \\
\text {x3}(t)\to \frac {1}{25} e^{-i t} \left (c_1 \left (e^{(2+i) t} (5 t+6)-(3+4 i) e^{2 i t}-(3-4 i)\right )+c_2 \left (2 e^{(2+i) t} (5 t-4)+(4-3 i) e^{2 i t}+(4+3 i)\right )+25 e^{(2+i) t} (c_4 t+c_3)\right ) \\
\text {x4}(t)\to \frac {1}{10} e^{-i t} \left (c_1 \left (-(1-2 i) e^{2 i t}+2 e^{(2+i) t}+(-1-2 i)\right )+c_2 \left (-(2+i) e^{2 i t}+4 e^{(2+i) t}+(-2+i)\right )+10 c_4 e^{(2+i) t}\right ) \\
\end{align*}
✓ Sympy. Time used: 0.194 (sec). Leaf size: 100
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__2(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) + Derivative(x__2(t), t),0),Eq(-x__1(t) - 2*x__3(t) - x__4(t) + Derivative(x__3(t), t),0),Eq(-x__2(t) - 2*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)
\[
\left [ x^{1}{\left (t \right )} = \left (C_{1} + 2 C_{2}\right ) \sin {\left (t \right )} + \left (2 C_{1} - C_{2}\right ) \cos {\left (t \right )}, \ x^{2}{\left (t \right )} = - \left (C_{1} + 2 C_{2}\right ) \cos {\left (t \right )} + \left (2 C_{1} - C_{2}\right ) \sin {\left (t \right )}, \ x^{3}{\left (t \right )} = C_{3} e^{2 t} + C_{4} t e^{2 t} + \left (\frac {2 C_{1}}{5} - \frac {4 C_{2}}{5}\right ) \sin {\left (t \right )} - \left (\frac {4 C_{1}}{5} + \frac {2 C_{2}}{5}\right ) \cos {\left (t \right )}, \ x^{4}{\left (t \right )} = - C_{1} \sin {\left (t \right )} + C_{2} \cos {\left (t \right )} + C_{4} e^{2 t}\right ]
\]