80.8.9 problem 13

Internal problem ID [21366]
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 : 13
Date solved : Sunday, October 12, 2025 at 05:51:29 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-7 x \left (t \right ) y \left (t \right )-a x \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-y \left (t \right )+4 x \left (t \right ) y \left (t \right )-a y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.394 (sec). Leaf size: 200

ode:=[diff(x(t),t) = 2*x(t)-7*x(t)*y(t)-a*x(t), diff(y(t),t) = -y(t)+4*x(t)*y(t)-a*y(t)]; 
dsolve(ode);
 

\begin{align*} [\{x \left (t \right ) = 0\}, \{y \left (t \right ) &= c_1 \,{\mathrm e}^{\left (-a -1\right ) t}\}] \\ \left [\left \{x \left (t \right ) &= \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}-\frac {1}{\left (\operatorname {LambertW}\left (-\frac {\textit {\_a}^{\frac {a}{a -2}} \textit {\_a}^{\frac {1}{a -2}} {\mathrm e}^{-\frac {c_1}{a -2}} {\mathrm e}^{-\frac {4 \textit {\_a}}{a -2}} {\mathrm e}^{-\frac {a}{a -2}} {\mathrm e}^{\frac {2}{a -2}}}{a -2}\right ) a -2 \operatorname {LambertW}\left (-\frac {\textit {\_a}^{\frac {a}{a -2}} \textit {\_a}^{\frac {1}{a -2}} {\mathrm e}^{-\frac {c_1}{a -2}} {\mathrm e}^{-\frac {4 \textit {\_a}}{a -2}} {\mathrm e}^{-\frac {a}{a -2}} {\mathrm e}^{\frac {2}{a -2}}}{a -2}\right )+a -2\right ) \textit {\_a}}d \textit {\_a} +t +c_2 \right )\right \}, \left \{y \left (t \right ) = \frac {-a x \left (t \right )-\frac {d}{d t}x \left (t \right )+2 x \left (t \right )}{7 x \left (t \right )}\right \}\right ] \\ \end{align*}

✗ Mathematica

ode={D[x[t],t]==2*x[t]-7*x[t]*y[t]-a*y[t],D[y[t],t]==-y[t]+4*x[t]*y[t]-a*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
t = symbols("t") 
a = symbols("a") 
x = Function("x") 
y = Function("y") 
ode=[Eq(a*y(t) + 7*x(t)*y(t) - 2*x(t) + Derivative(x(t), t),0),Eq(a*y(t) - 4*x(t)*y(t) + y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

Timed Out