69.29.2 problem 803

Internal problem ID [18543]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 22. Integration of homogeneous linear systems with constant coefficients. Eulers method. Exercises page 230
Problem number : 803
Date solved : Thursday, October 02, 2025 at 03:14:44 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.113 (sec). Leaf size: 24

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 60

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

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

✓ Sympy. Time used: 0.036 (sec). Leaf size: 20

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

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