64.16.17 problem 17

Internal problem ID [13549]
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 : 17
Date solved : Monday, March 31, 2025 at 08:01:04 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 )-x \left (t \right )-y \left (t \right )&=1\\ \frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )+2 x \left (t \right )-y \left (t \right )&=t \end{align*}

✓ Maple. Time used: 0.157 (sec). Leaf size: 34

ode:=[2*diff(x(t),t)+diff(y(t),t)-x(t)-y(t) = 1, diff(x(t),t)+diff(y(t),t)+2*x(t)-y(t) = t]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= \frac {t}{3}-\frac {2}{9}+{\mathrm e}^{3 t} c_2 \\ y \left (t \right ) &= -\frac {4}{9}-\frac {t}{3}-\frac {5 \,{\mathrm e}^{3 t} c_2}{2}+c_1 \,{\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.183 (sec). Leaf size: 127

ode={2*D[x[t],t]+D[y[t],t]-x[t]-y[t]==1,D[x[t],t]+D[y[t],t]+2*x[t]-y[t]==t}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{3 t} \left (\int _1^t-e^{-3 K[1]} (K[1]-1)dK[1]+c_1\right ) \\ y(t)\to \frac {1}{2} e^t \left (-5 \left (e^{2 t}-1\right ) \int _1^t-e^{-3 K[1]} (K[1]-1)dK[1]+2 \int _1^t\frac {1}{2} e^{-3 K[2]} \left (5 (K[2]-1)-e^{2 K[2]} (K[2]-3)\right )dK[2]-5 c_1 e^{2 t}+5 c_1+2 c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.183 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) - y(t) + 2*Derivative(x(t), t) + Derivative(y(t), t) - 1,0),Eq(-t + 2*x(t) - y(t) + Derivative(x(t), t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {2 C_{1} e^{3 t}}{5} + \frac {t}{3} - \frac {2}{9}, \ y{\left (t \right )} = C_{1} e^{3 t} + C_{2} e^{t} - \frac {t}{3} - \frac {4}{9}\right ] \]