69.29.1 problem 802

Internal problem ID [18542]
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 : 802
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 )&=8 y \left (t \right )-x \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.105 (sec). Leaf size: 35

ode:=[diff(x(t),t) = 8*y(t)-x(t), diff(y(t),t) = x(t)+y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 72

ode={D[x[t],t]==8*y[t]-x[t],D[y[t],t]==x[t]+y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{3} e^{-3 t} \left (c_1 \left (e^{6 t}+2\right )+4 c_2 \left (e^{6 t}-1\right )\right )\\ y(t)&\to \frac {1}{6} e^{-3 t} \left (c_1 \left (e^{6 t}-1\right )+2 c_2 \left (2 e^{6 t}+1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.054 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(x(t) - 8*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)
 

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