83.13.1 problem A(1)

Internal problem ID [22009]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter IX. System of equations. Ex. XVII at page 154
Problem number : A(1)
Date solved : Thursday, October 02, 2025 at 08:21:38 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+x \left (t \right )+\frac {d}{d t}y \left (t \right )+y \left (t \right )&=0\\ \frac {d}{d t}x \left (t \right )-\frac {d}{d t}y \left (t \right )-y \left (t \right )&=t \end{align*}

✓ Maple. Time used: 0.053 (sec). Leaf size: 34

ode:=[diff(x(t),t)+x(t)+diff(y(t),t)+y(t) = 0, diff(x(t),t)-diff(y(t),t)-y(t) = t]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 51

ode={D[x[t],t]+x[t]+D[y[t],t]+y[t]==0,D[x[t],t]-D[y[t],t]-y[t]==t}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to t+c_1 e^{-t/2}-2\\ y(t)&\to e^{-t} \left (-e^t (t-2)-c_1 e^{t/2}+c_1+c_2\right ) \end{align*}

✓ Sympy. Time used: 0.082 (sec). Leaf size: 29

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

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