28.4.13 problem 7.13
Internal
problem
ID
[4545]
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
:
7.13
Date
solved
:
Tuesday, March 04, 2025 at 06:52:10 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )+2 x \left (t \right )+5 y \left (t \right )&=0\\ -x \left (t \right )+\frac {d}{d t}y \left (t \right )-2 y \left (t \right )&=\sin \left (2 t \right ) \end{align*}
✓ Maple. Time used: 0.171 (sec). Leaf size: 55
ode:=[diff(x(t),t)+2*x(t)+5*y(t) = 0, -x(t)+diff(y(t),t)-2*y(t) = sin(2*t)];
dsolve(ode);
\begin{align*}
x &= \sin \left (t \right ) c_{2} +\cos \left (t \right ) c_{1} +\frac {5 \sin \left (2 t \right )}{3} \\
y &= -\frac {c_{2} \cos \left (t \right )}{5}+\frac {c_{1} \sin \left (t \right )}{5}-\frac {2 \cos \left (2 t \right )}{3}-\frac {2 \sin \left (t \right ) c_{2}}{5}-\frac {2 \cos \left (t \right ) c_{1}}{5}-\frac {2 \sin \left (2 t \right )}{3} \\
\end{align*}
✓ Mathematica. Time used: 0.014 (sec). Leaf size: 67
ode={D[x[t],t]+2*x[t]+5*y[t]==0,-x[t]+D[y[t],t]-2*y[t]==Sin[2*t]};
ic={};
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {5}{3} \sin (2 t)+c_1 \cos (t)-2 c_1 \sin (t)-5 c_2 \sin (t) \\
y(t)\to -\frac {2}{3} \sin (2 t)-\frac {2}{3} \cos (2 t)+c_2 \cos (t)+c_1 \sin (t)+2 c_2 \sin (t) \\
\end{align*}
✓ Sympy. Time used: 0.268 (sec). Leaf size: 112
from sympy import *
t = symbols("t")
x = Function("x")
y = Function("y")
ode=[Eq(2*x(t) + 5*y(t) + Derivative(x(t), t),0),Eq(-x(t) - 2*y(t) - sin(2*t) + Derivative(y(t), t),0)]
ics = {}
dsolve(ode,func=[x(t),y(t)],ics=ics)
\[
\left [ x{\left (t \right )} = - \left (C_{1} - 2 C_{2}\right ) \sin {\left (t \right )} - \left (2 C_{1} + C_{2}\right ) \cos {\left (t \right )} + \frac {5 \sin ^{2}{\left (t \right )} \sin {\left (2 t \right )}}{3} + \frac {5 \sin {\left (2 t \right )} \cos ^{2}{\left (t \right )}}{3}, \ y{\left (t \right )} = C_{1} \cos {\left (t \right )} - C_{2} \sin {\left (t \right )} - \frac {2 \sin ^{2}{\left (t \right )} \sin {\left (2 t \right )}}{3} - \frac {2 \sin ^{2}{\left (t \right )} \cos {\left (2 t \right )}}{3} - \frac {2 \sin {\left (2 t \right )} \cos ^{2}{\left (t \right )}}{3} - \frac {2 \cos ^{2}{\left (t \right )} \cos {\left (2 t \right )}}{3}\right ]
\]