80.8.6 problem 10

Internal problem ID [21363]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 8. Qualitative analysis of 2 by 2 systems and nonlinear second order equations. Excercise 8.5 at page 184
Problem number : 10
Date solved : Thursday, October 02, 2025 at 07:28:47 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=x-6 y \left (t \right )\\ y^{\prime }\left (t \right )&=-2 x-y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=1 \\ y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.064 (sec). Leaf size: 106

ode:=[diff(x(t),t) = x(t)-6*y(t), diff(y(t),t) = -2*x(t)-y(t)]; 
ic:=[x(0) = 1, y(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= \left (\frac {1}{2}+\frac {\sqrt {13}}{26}\right ) {\mathrm e}^{\sqrt {13}\, t}+\left (\frac {1}{2}-\frac {\sqrt {13}}{26}\right ) {\mathrm e}^{-\sqrt {13}\, t} \\ y \left (t \right ) &= -\frac {\left (\frac {1}{2}+\frac {\sqrt {13}}{26}\right ) \sqrt {13}\, {\mathrm e}^{\sqrt {13}\, t}}{6}+\frac {\left (\frac {1}{2}-\frac {\sqrt {13}}{26}\right ) \sqrt {13}\, {\mathrm e}^{-\sqrt {13}\, t}}{6}+\frac {\left (\frac {1}{2}+\frac {\sqrt {13}}{26}\right ) {\mathrm e}^{\sqrt {13}\, t}}{6}+\frac {\left (\frac {1}{2}-\frac {\sqrt {13}}{26}\right ) {\mathrm e}^{-\sqrt {13}\, t}}{6} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 78

ode={D[x[t],t]==x[t]-6*y[t],D[y[t],t]==-2*x[t]-y[t]}; 
ic={x[0]==1,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.136 (sec). Leaf size: 68

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) + 6*y(t) + Derivative(x(t), t),0),Eq(2*x(t) + y(t) + Derivative(y(t), t),0)] 
ics = {x(0): 1, y(0): 0} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {\left (\sqrt {13} + 13\right ) e^{\sqrt {13} t}}{26} + \frac {\left (13 - \sqrt {13}\right ) e^{- \sqrt {13} t}}{26}, \ y{\left (t \right )} = - \frac {\sqrt {13} e^{\sqrt {13} t}}{13} + \frac {\sqrt {13} e^{- \sqrt {13} t}}{13}\right ] \]