28.4.52 problem 7.52
Internal
problem
ID
[4584]
Book
:
Differential
equations
for
engineers
by
Wei-Chau
XIE,
Cambridge
Press
2010
Section
:
Chapter
7.
Systems
of
linear
differential
equations.
Problems
at
page
351
Problem
number
:
7.52
Date
solved
:
Tuesday, March 04, 2025 at 06:55:59 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )-x_{2} \left (t \right )+2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )+2 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-2 x_{1} \left (t \right )+x_{2} \left (t \right )-x_{3} \left (t \right )+4 \sin \left (t \right ) \end{align*}
✓ Maple. Time used: 0.370 (sec). Leaf size: 98
ode:=[diff(x__1(t),t) = 2*x__1(t)-x__2(t)+2*x__3(t), diff(x__2(t),t) = x__1(t)+2*x__3(t), diff(x__3(t),t) = -2*x__1(t)+x__2(t)-x__3(t)+4*sin(t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= \cos \left (t \right ) c_{1} -4 \cos \left (t \right ) t +c_3 \sin \left (t \right )-4 \cos \left (t \right )+4 \sin \left (t \right ) \\
x_{2} \left (t \right ) &= \cos \left (t \right ) c_{1} -4 \cos \left (t \right ) t +c_3 \sin \left (t \right )+c_{2} {\mathrm e}^{t}-4 \cos \left (t \right )+4 \sin \left (t \right ) \\
x_{3} \left (t \right ) &= -\frac {\cos \left (t \right ) c_{1}}{2}+\frac {\cos \left (t \right ) c_3}{2}+2 \cos \left (t \right ) t -\frac {c_{1} \sin \left (t \right )}{2}-\frac {c_3 \sin \left (t \right )}{2}+2 \sin \left (t \right ) t +\frac {c_{2} {\mathrm e}^{t}}{2}+2 \cos \left (t \right ) \\
\end{align*}
✓ Mathematica. Time used: 0.035 (sec). Leaf size: 125
ode={D[x1[t],t]==2*x1[t]-x2[t]+2*x3[t],D[x2[t],t]==x1[t]+2*x3[t],D[x3[t],t]==-2*x1[t]+x2[t]-x3[t]+4*Sin[t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {x1}(t)\to (-4 t+c_1) \cos (t)+(2 c_1-c_2+2 c_3) \sin (t) \\
\text {x2}(t)\to (c_2-c_1) e^t+(-4 t+c_1) \cos (t)+(2 c_1-c_2+2 c_3) \sin (t) \\
\text {x3}(t)\to \frac {1}{2} \left ((c_2-c_1) e^t+(4 t-4+c_1-c_2+2 c_3) \cos (t)+(4 t-3 c_1+c_2-2 c_3) \sin (t)\right ) \\
\end{align*}
✓ Sympy. Time used: 0.259 (sec). Leaf size: 187
from sympy import *
t = symbols("t")
x__1 = Function("x__1")
x__2 = Function("x__2")
x__3 = Function("x__3")
ode=[Eq(-2*x__1(t) + x__2(t) - 2*x__3(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) - 2*x__3(t) + Derivative(x__2(t), t),0),Eq(2*x__1(t) - x__2(t) + x__3(t) - 4*sin(t) + Derivative(x__3(t), t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = - 4 t \sin ^{2}{\left (t \right )} \cos {\left (t \right )} - 4 t \cos ^{3}{\left (t \right )} - \left (C_{1} - C_{2}\right ) \cos {\left (t \right )} + \left (C_{1} + C_{2}\right ) \sin {\left (t \right )} + 4 \sin ^{3}{\left (t \right )} + 4 \sin {\left (t \right )} \cos ^{2}{\left (t \right )}, \ x^{2}{\left (t \right )} = 2 C_{3} e^{t} - 4 t \sin ^{2}{\left (t \right )} \cos {\left (t \right )} - 4 t \cos ^{3}{\left (t \right )} - \left (C_{1} - C_{2}\right ) \cos {\left (t \right )} + \left (C_{1} + C_{2}\right ) \sin {\left (t \right )} + 4 \sin ^{3}{\left (t \right )} + 4 \sin {\left (t \right )} \cos ^{2}{\left (t \right )}, \ x^{3}{\left (t \right )} = C_{1} \cos {\left (t \right )} - C_{2} \sin {\left (t \right )} + C_{3} e^{t} + 2 t \sin ^{3}{\left (t \right )} + 2 t \sin ^{2}{\left (t \right )} \cos {\left (t \right )} + 2 t \sin {\left (t \right )} \cos ^{2}{\left (t \right )} + 2 t \cos ^{3}{\left (t \right )} - 2 \sin ^{3}{\left (t \right )} - 2 \sin {\left (t \right )} \cos ^{2}{\left (t \right )}\right ]
\]