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 : Tuesday, January 28, 2025 at 02:39:22 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.041 (sec). Leaf size: 85

dsolve([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)],singsol=all)
 

\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 \ln \left ({\mathrm e}^{t}-1\right )+6 \,{\mathrm e}^{-t}+3 c_{1} +4 c_{2} -6}{2 \,{\mathrm e}^{t}-2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 76

DSolve[{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)},{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*}