82.6.8 problem 33-50

Internal problem ID [21855]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 33. Systems of ordinary differential equations. Page 1059
Problem number : 33-50
Date solved : Thursday, October 02, 2025 at 08:02:57 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 )&=9 x \left (t \right )+y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.039 (sec). Leaf size: 35

ode:=[diff(x(t),t) = x(t)+y(t), diff(y(t),t) = 9*x(t)+y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 70

ode={D[x[t],t]==x[t]+y[t],D[y[t],t]==9*x[t]+y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.264 (sec). Leaf size: 105

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) - y(t) + z(t) + Derivative(x(t), t),0),Eq(-9*x(t) - y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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