87.31.20 problem 27

Internal problem ID [23936]
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 : 27
Date solved : Thursday, October 02, 2025 at 09:46:38 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.043 (sec). Leaf size: 81

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

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{\left (\sqrt {5}-1\right ) t}+c_2 \,{\mathrm e}^{-\left (\sqrt {5}+1\right ) t} \\ y \left (t \right ) &= c_1 \,{\mathrm e}^{\left (\sqrt {5}-1\right ) t} \sqrt {5}-c_2 \,{\mathrm e}^{-\left (\sqrt {5}+1\right ) t} \sqrt {5}-2 c_1 \,{\mathrm e}^{\left (\sqrt {5}-1\right ) t}-2 c_2 \,{\mathrm e}^{-\left (\sqrt {5}+1\right ) t} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 147

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

\begin{align*} x(t)&\to \frac {1}{10} e^{-\left (\left (1+\sqrt {5}\right ) t\right )} \left (c_1 \left (\left (5+2 \sqrt {5}\right ) e^{2 \sqrt {5} t}+5-2 \sqrt {5}\right )+\sqrt {5} c_2 \left (e^{2 \sqrt {5} t}-1\right )\right )\\ y(t)&\to \frac {1}{10} e^{-\left (\left (1+\sqrt {5}\right ) t\right )} \left (\sqrt {5} c_1 \left (e^{2 \sqrt {5} t}-1\right )-c_2 \left (\left (2 \sqrt {5}-5\right ) e^{2 \sqrt {5} t}-5-2 \sqrt {5}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.114 (sec). Leaf size: 65

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

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