82.7.4 problem 34-6

Internal problem ID [21860]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 34. Simulataneous linear differential equations. Page 1118
Problem number : 34-6
Date solved : Thursday, October 02, 2025 at 08:03:00 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )+2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-5 x \left (t \right )+y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.059 (sec). Leaf size: 58

ode:=[diff(x(t),t) = 3*x(t)+2*y(t), diff(y(t),t) = -5*x(t)+y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 69

ode={D[x[t],t]==3*x[t]+2*y[t],D[y[t],t]==-5*x[t]+y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{3} e^{2 t} (3 c_1 \cos (3 t)+(c_1+2 c_2) \sin (3 t))\\ y(t)&\to \frac {1}{3} e^{2 t} (3 c_2 \cos (3 t)-(5 c_1+c_2) \sin (3 t)) \end{align*}

✓ Sympy. Time used: 0.070 (sec). Leaf size: 65

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-3*x(t) - 2*y(t) + Derivative(x(t), t),0),Eq(5*x(t) - y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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