80.8.4 problem 6

Internal problem ID [21361]
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 : 6
Date solved : Thursday, October 02, 2025 at 07:28:46 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.065 (sec). Leaf size: 98

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

\begin{align*} x \left (t \right ) &= c_1 \sin \left (\sqrt {2 c -1}\, t \right )+c_2 \cos \left (\sqrt {2 c -1}\, t \right ) \\ y \left (t \right ) &= c_1 \sqrt {2 c -1}\, \cos \left (\sqrt {2 c -1}\, t \right )-c_2 \sqrt {2 c -1}\, \sin \left (\sqrt {2 c -1}\, t \right )-c_1 \sin \left (\sqrt {2 c -1}\, t \right )-c_2 \cos \left (\sqrt {2 c -1}\, t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 184

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

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

✓ Sympy. Time used: 0.137 (sec). Leaf size: 78

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

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