82.7.2 problem 34-3

Internal problem ID [21858]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 34. Simulataneous linear differential equations. Page 1118
Problem number : 34-3
Date solved : Thursday, October 02, 2025 at 08:02:59 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.108 (sec). Leaf size: 34

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

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

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 81

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

\begin{align*} x(t)&\to \frac {1}{2} \left (2 \sin (t)+2 \cos (t)+c_1 e^{-t}+c_1 e^t+c_2 e^{-t}-c_2 e^t\right )\\ y(t)&\to \frac {1}{2} e^{-t} \left (c_1 \left (-e^{2 t}\right )+c_2 e^{2 t}+c_1+c_2\right ) \end{align*}

✓ Sympy. Time used: 0.162 (sec). Leaf size: 31

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

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