75.30.4 problem 813

Internal problem ID [17187]
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
Date solved : Friday, March 14, 2025 at 04:49:30 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}\\ \frac {d}{d t}y \left (t \right )&=6 x \left (t \right )+3 y \left (t \right )-\frac {3}{{\mathrm e}^{t}-1} \end{align*}

✓ Maple. Time used: 0.222 (sec). Leaf size: 85

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

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 76

ode={D[x[t],t]==-4*x[t]-2*y[t]+2/(Exp[t]-1),D[y[t],t]==6*x[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: 0.253 (sec). Leaf size: 63

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)
 

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