58.16.15 problem 15

Internal problem ID [14899]
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 : 15
Date solved : Thursday, October 02, 2025 at 09:56:23 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.168 (sec). Leaf size: 44

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-2 t} c_2 +{\mathrm e}^{t} c_1 -t \,{\mathrm e}^{t} \\ y \left (t \right ) &= -\frac {{\mathrm e}^{-2 t} c_2}{3}-{\mathrm e}^{t} c_1 +\frac {{\mathrm e}^{t}}{3}+t \,{\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 76

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

\begin{align*} x(t)&\to \frac {3}{2} (c_1+c_2) e^{-2 t}-\frac {1}{2} e^t (2 t-1+c_1+3 c_2)\\ y(t)&\to \frac {1}{6} e^t (6 t-1+3 c_1+9 c_2)-\frac {1}{2} (c_1+c_2) e^{-2 t} \end{align*}

✓ Sympy. Time used: 0.104 (sec). Leaf size: 48

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

\[ \left [ x{\left (t \right )} = - 3 C_{2} e^{- 2 t} - t e^{t} - \left (C_{1} - \frac {1}{2}\right ) e^{t}, \ y{\left (t \right )} = C_{2} e^{- 2 t} + t e^{t} + \left (C_{1} - \frac {1}{6}\right ) e^{t}\right ] \]