28.4.18 problem 7.18

Internal problem ID [4550]
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 : 7.18
Date solved : Sunday, March 30, 2025 at 03:26:04 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.254 (sec). Leaf size: 85

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

\begin{align*} x \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 \\ y \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: 0.023 (sec). Leaf size: 76

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

\begin{align*} x(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 ) \\ y(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: 0.244 (sec). Leaf size: 46

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

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