87.31.22 problem 29

Internal problem ID [23938]
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 : 29
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 )&=-2 x \left (t \right )-3 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-3 x \left (t \right )+2 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.041 (sec). Leaf size: 70

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

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{\sqrt {13}\, t}+c_2 \,{\mathrm e}^{-\sqrt {13}\, t} \\ y \left (t \right ) &= -\frac {c_1 \sqrt {13}\, {\mathrm e}^{\sqrt {13}\, t}}{3}+\frac {c_2 \sqrt {13}\, {\mathrm e}^{-\sqrt {13}\, t}}{3}-\frac {2 c_1 \,{\mathrm e}^{\sqrt {13}\, t}}{3}-\frac {2 c_2 \,{\mathrm e}^{-\sqrt {13}\, t}}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 144

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

\begin{align*} x(t)&\to \frac {1}{26} e^{-\sqrt {13} t} \left (c_1 \left (\left (13-2 \sqrt {13}\right ) e^{2 \sqrt {13} t}+13+2 \sqrt {13}\right )-3 \sqrt {13} c_2 \left (e^{2 \sqrt {13} t}-1\right )\right )\\ y(t)&\to \frac {1}{26} e^{-\sqrt {13} t} \left (c_2 \left (\left (13+2 \sqrt {13}\right ) e^{2 \sqrt {13} t}+13-2 \sqrt {13}\right )-3 \sqrt {13} c_1 \left (e^{2 \sqrt {13} t}-1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.113 (sec). Leaf size: 61

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

\[ \left [ x{\left (t \right )} = \frac {C_{1} \left (2 - \sqrt {13}\right ) e^{\sqrt {13} t}}{3} + \frac {C_{2} \left (2 + \sqrt {13}\right ) e^{- \sqrt {13} t}}{3}, \ y{\left (t \right )} = C_{1} e^{\sqrt {13} t} + C_{2} e^{- \sqrt {13} t}\right ] \]