87.31.21 problem 28

Internal problem ID [23937]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 8. Nonlinear differential equations and systems. Exercise at page 321
Problem number : 28
Date solved : Thursday, October 02, 2025 at 09:46:39 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 )&=2 x \left (t \right )+3 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.062 (sec). Leaf size: 45

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 51

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

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

✓ Sympy. Time used: 0.066 (sec). Leaf size: 53

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(-2*x(t) - 3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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