82.6.6 problem 33-39

Internal problem ID [21853]
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 : 33-39
Date solved : Thursday, October 02, 2025 at 08:02:56 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-5 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )-4 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.060 (sec). Leaf size: 47

ode:=[diff(x(t),t) = 2*x(t)-5*y(t), diff(y(t),t) = 2*x(t)-4*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \left (\sin \left (t \right ) c_1 +\cos \left (t \right ) c_2 \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (3 \sin \left (t \right ) c_1 +\sin \left (t \right ) c_2 -\cos \left (t \right ) c_1 +3 \cos \left (t \right ) c_2 \right )}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 56

ode={D[x[t],t]==2*x[t]-5*y[t],D[y[t],t]==2*x[t]-4*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{-t} (c_1 \cos (t)+(3 c_1-5 c_2) \sin (t))\\ y(t)&\to e^{-t} (c_2 \cos (t)+(2 c_1-3 c_2) \sin (t)) \end{align*}

✓ Sympy. Time used: 0.062 (sec). Leaf size: 53

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t) + 5*y(t) + Derivative(x(t), t),0),Eq(-2*x(t) + 4*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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