28.4.42 problem 7.42

Internal problem ID [4574]
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 : 7.42
Date solved : Sunday, March 30, 2025 at 03:26:38 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-4 x_{1} \left (t \right )-2 x_{2} \left (t \right )+\frac {2}{{\mathrm e}^{t}-1}\\ \frac {d}{d t}x_{2} \left (t \right )&=6 x_{1} \left (t \right )+3 x_{2} \left (t \right )-\frac {3}{{\mathrm e}^{t}-1} \end{align*}

✓ Maple. Time used: 0.239 (sec). Leaf size: 85

ode:=[diff(x__1(t),t) = -4*x__1(t)-2*x__2(t)+2/(exp(t)-1), diff(x__2(t),t) = 6*x__1(t)+3*x__2(t)-3/(exp(t)-1)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= 2 \,{\mathrm e}^{-t} \ln \left ({\mathrm e}^{t}-1\right )-{\mathrm e}^{-t} c_1 +2 \,{\mathrm e}^{-t}+c_2 \\ x_{2} \left (t \right ) &= \frac {6 \,{\mathrm e}^{-t} \ln \left ({\mathrm e}^{t}-1\right )-3 \,{\mathrm e}^{-t} c_1 -4 c_2 \,{\mathrm e}^{t}+6 \,{\mathrm e}^{-t}-6 \ln \left ({\mathrm e}^{t}-1\right )+3 c_1 +4 c_2 -6}{2 \,{\mathrm e}^{t}-2} \\ \end{align*}

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 76

ode={D[x1[t],t]==-4*x1[t]-2*x2[t]+2/(Exp[t]-1),D[x2[t],t]==6*x1[t]+3*x2[t]-3/(Exp[t]-1)}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{-t} \left (2 \log \left (e^t-1\right )+c_1 \left (4-3 e^t\right )-2 c_2 \left (e^t-1\right )\right ) \\ \text {x2}(t)\to e^{-t} \left (-3 \log \left (e^t-1\right )+6 c_1 \left (e^t-1\right )+c_2 \left (4 e^t-3\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.264 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(4*x__1(t) + 2*x__2(t) + Derivative(x__1(t), t) - 2/(exp(t) - 1),0),Eq(-6*x__1(t) - 3*x__2(t) + Derivative(x__2(t), t) + 3/(exp(t) - 1),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - \frac {C_{1}}{2} - \frac {2 C_{2} e^{- t}}{3} + 2 e^{- t} \log {\left (e^{t} - 1 \right )}, \ x^{2}{\left (t \right )} = C_{1} + C_{2} e^{- t} - 3 e^{- t} \log {\left (e^{t} - 1 \right )}\right ] \]