58.16.6 problem 6

Internal problem ID [14890]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 7, Systems of linear differential equations. Section 7.1. Exercises page 277
Problem number : 6
Date solved : Thursday, October 02, 2025 at 09:56:15 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.172 (sec). Leaf size: 41

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

\begin{align*} x \left (t \right ) &= t -1-\frac {{\mathrm e}^{t}}{2}+{\mathrm e}^{-t} c_2 \\ y \left (t \right ) &= -\frac {5 \,{\mathrm e}^{-t} c_2}{2}-4 t +1+c_1 \,{\mathrm e}^{t}-\frac {{\mathrm e}^{t} t}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.623 (sec). Leaf size: 72

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

\begin{align*} x(t)&\to -\frac {t}{7}+\frac {e^t}{6}+c_1 e^{7 t}-\frac {1}{49}\\ y(t)&\to -\frac {4 t}{7}-\frac {11}{6} c_1 e^{7 t}+\frac {1}{36} e^t (6 t-11+66 c_1+36 c_2)-\frac {39}{49} \end{align*}

✓ Sympy. Time used: 0.113 (sec). Leaf size: 46

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

\[ \left [ x{\left (t \right )} = - \frac {2 C_{1} e^{- t}}{5} + t - \frac {e^{t}}{2} - 1, \ y{\left (t \right )} = C_{1} e^{- t} - \frac {t e^{t}}{2} - 4 t + \left (C_{2} + \frac {5}{4}\right ) e^{t} + 1\right ] \]