28.4.44 problem 7.44
Internal
problem
ID
[4576]
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
:
7.44
Date
solved
:
Tuesday, March 04, 2025 at 06:52:41 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-x_{1} \left (t \right )-5 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )+\frac {4}{\sin \left (2 t \right )} \end{align*}
✓ Maple. Time used: 0.345 (sec). Leaf size: 160
ode:=[diff(x__1(t),t) = -x__1(t)-5*x__2(t), diff(x__2(t),t) = x__1(t)+x__2(t)+4/sin(2*t)];
dsolve(ode);
\begin{align*}
x_{1} \left (t \right ) &= -10 \sin \left (2 t \right ) \ln \left (\sin \left (t \right )\right )+5 \sin \left (2 t \right ) \ln \left (\tan \left (t \right )\right )+c_{2} \sin \left (2 t \right )+c_{1} \cos \left (2 t \right )+10 t \cos \left (2 t \right ) \\
x_{2} \left (t \right ) &= -\tan \left (t \right ) \sin \left (2 t \right )+2 \sin \left (2 t \right ) \ln \left (\sin \left (t \right )\right )-\sin \left (2 t \right ) \ln \left (\tan \left (t \right )\right )+\frac {2 c_{1} \sin \left (2 t \right )}{5}-\frac {c_{2} \sin \left (2 t \right )}{5}+4 t \sin \left (2 t \right )+4 \cos \left (2 t \right ) \ln \left (\sin \left (t \right )\right )-2 \cos \left (2 t \right ) \ln \left (\tan \left (t \right )\right )-\frac {c_{1} \cos \left (2 t \right )}{5}-\frac {2 c_{2} \cos \left (2 t \right )}{5}-2 t \cos \left (2 t \right )+\frac {2 \cos \left (t \right ) \sin \left (2 t \right )}{\sin \left (t \right )}-2 \cos \left (2 t \right )-\frac {\sin \left (2 t \right )}{\tan \left (t \right )} \\
\end{align*}
✓ Mathematica. Time used: 0.014 (sec). Leaf size: 87
ode={D[x1[t],t]==-x1[t]-5*x2[t],D[x2[t],t]==x1[t]+x2[t]+4/Sin[2*t]};
ic={};
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
\begin{align*}
\text {x1}(t)\to (10 t+c_1) \cos (2 t)-\frac {1}{2} \sin (2 t) (10 \log (\sin (2 t))+c_1+5 c_2) \\
\text {x2}(t)\to \frac {1}{2} (\sin (2 t) (8 t+2 \log (\sin (2 t))+c_1+c_2)+\cos (2 t) (-4 t+4 \log (\sin (2 t))+2 c_2)) \\
\end{align*}
✓ Sympy. Time used: 0.148 (sec). Leaf size: 102
from sympy import *
t = symbols("t")
x__1 = Function("x__1")
x__2 = Function("x__2")
ode=[Eq(x__1(t) + 5*x__2(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) - x__2(t) + Derivative(x__2(t), t) - 4/sin(2*t),0)]
ics = {}
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
\[
\left [ x^{1}{\left (t \right )} = 10 t \cos {\left (2 t \right )} - \left (C_{1} + 2 C_{2}\right ) \cos {\left (2 t \right )} - \left (2 C_{1} - C_{2}\right ) \sin {\left (2 t \right )} - 5 \log {\left (\sin {\left (2 t \right )} \right )} \sin {\left (2 t \right )}, \ x^{2}{\left (t \right )} = C_{1} \cos {\left (2 t \right )} - C_{2} \sin {\left (2 t \right )} + 4 t \sin {\left (2 t \right )} - 2 t \cos {\left (2 t \right )} + \log {\left (\sin {\left (2 t \right )} \right )} \sin {\left (2 t \right )} + 2 \log {\left (\sin {\left (2 t \right )} \right )} \cos {\left (2 t \right )}\right ]
\]