64.16.1 problem 1

Internal problem ID [13533]
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 : 1
Date solved : Monday, March 31, 2025 at 08:00:38 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.132 (sec). Leaf size: 31

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

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{-2 t} \\ y \left (t \right ) &= \frac {{\mathrm e}^{4 t}}{3}-\frac {{\mathrm e}^{t}}{3}-\frac {2 c_1 \,{\mathrm e}^{-2 t}}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.055 (sec). Leaf size: 52

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

\begin{align*} x(t)\to \frac {1}{12} (3+4 c_1) e^{-2 t} \\ y(t)\to \frac {1}{18} e^{-2 t} \left (-6 e^{3 t}+6 e^{6 t}-3-4 c_1\right ) \\ \end{align*}

✗ Sympy

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