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 : Tuesday, January 28, 2025 at 02:39:20 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.090 (sec). Leaf size: 85

dsolve([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)],singsol=all)
 

\begin{align*} x &= 2 \,{\mathrm e}^{-t} \ln \left ({\mathrm e}^{t}-1\right )-{\mathrm e}^{-t} c_{1} +2 \,{\mathrm e}^{-t}+c_{2} \\ y &= \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: 0.023 (sec). Leaf size: 76

DSolve[{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)},{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*}