82.6.9 problem 33-51

Internal problem ID [21856]
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 : 33-51
Date solved : Thursday, October 02, 2025 at 08:02:58 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=4 x \left (t \right )+3 y \left (t \right ) \end{align*}

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

ode:=[diff(x(t),t) = x(t)+2*y(t), diff(y(t),t) = 4*x(t)+3*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{-t}+c_2 \,{\mathrm e}^{5 t} \\ y \left (t \right ) &= -c_1 \,{\mathrm e}^{-t}+2 c_2 \,{\mathrm e}^{5 t} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 71

ode={D[x[t],t]==x[t]+2*y[t],D[y[t],t]==4*x[t]+3*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.241 (sec). Leaf size: 95

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) - 2*y(t) + z(t) + Derivative(x(t), t),0),Eq(-4*x(t) - 3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - C_{1} e^{- t} + \frac {C_{2} e^{5 t}}{2} - \frac {e^{5 t} \int z{\left (t \right )} e^{- 5 t}\, dt}{3} - \frac {2 e^{- t} \int z{\left (t \right )} e^{t}\, dt}{3}, \ y{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{5 t} - \frac {2 e^{5 t} \int z{\left (t \right )} e^{- 5 t}\, dt}{3} + \frac {2 e^{- t} \int z{\left (t \right )} e^{t}\, dt}{3}\right ] \]