Internal problem ID [15535]
Internal file name [OUTPUT/15536_Friday_May_10_2024_09_47_53_PM_96363/index.tex]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 3 (Systems of differential equations). Section 23. Methods of integrating
nonhomogeneous linear systems with constant coefficients. Exercises page 234
Problem number: 813.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "system of linear ODEs" Unable to solve or complete the solution.
Solve \begin {align*} x^{\prime }\left (t \right )&=-\frac {4 \,{\mathrm e}^{t} x \left (t \right )}{{\mathrm e}^{t}-1}-\frac {2 \,{\mathrm e}^{t} y \left (t \right )}{{\mathrm e}^{t}-1}+\frac {4 x \left (t \right )}{{\mathrm e}^{t}-1}+\frac {2 y \left (t \right )}{{\mathrm e}^{t}-1}+\frac {2}{{\mathrm e}^{t}-1}\\ y^{\prime }\left (t \right )&=\frac {6 \,{\mathrm e}^{t} x \left (t \right )}{{\mathrm e}^{t}-1}+\frac {3 \,{\mathrm e}^{t} y \left (t \right )}{{\mathrm e}^{t}-1}-\frac {6 x \left (t \right )}{{\mathrm e}^{t}-1}-\frac {3 y \left (t \right )}{{\mathrm e}^{t}-1}-\frac {3}{{\mathrm e}^{t}-1} \end {align*}
Does not currently support non autonomous system of first order linear differential equations. The following is the phase plot
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 86
dsolve([diff(x(t),t)=-4*x(t)-2*y(t)+2/(exp(t)-1),diff(y(t),t)=6*x(t)+3*y(t)-3/(exp(t)-1)],singsol=all)
\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 )-4 c_{2} {\mathrm e}^{t}-3 \,{\mathrm e}^{-t} c_{1} -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[{x'[t]==-4*x[t]-2*y[t]+2/(Exp[t]-1),y'[t]==6*x[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*}