22.4.13 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, September 30, 2025 at 07:34:11 AM
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: 0.268 (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 \left (t \right ) &= \sin \left (t \right ) c_2 +\cos \left (t \right ) c_1 +\frac {5 \sin \left (2 t \right )}{3} \\ y \left (t \right ) &= -\frac {\cos \left (t \right ) c_2}{5}+\frac {\sin \left (t \right ) c_1}{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: 0.01 (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: 0.163 (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)
 

\[ \left [ x{\left (t \right )} = - \left (C_{1} - 2 C_{2}\right ) \sin {\left (t \right )} - \left (2 C_{1} + C_{2}\right ) \cos {\left (t \right )} + \frac {5 \sin ^{2}{\left (t \right )} \sin {\left (2 t \right )}}{3} + \frac {5 \sin {\left (2 t \right )} \cos ^{2}{\left (t \right )}}{3}, \ y{\left (t \right )} = C_{1} \cos {\left (t \right )} - C_{2} \sin {\left (t \right )} - \frac {2 \sin ^{2}{\left (t \right )} \sin {\left (2 t \right )}}{3} - \frac {2 \sin ^{2}{\left (t \right )} \cos {\left (2 t \right )}}{3} - \frac {2 \sin {\left (2 t \right )} \cos ^{2}{\left (t \right )}}{3} - \frac {2 \cos ^{2}{\left (t \right )} \cos {\left (2 t \right )}}{3}\right ] \]