80.8.10 problem 14

Internal problem ID [21367]
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 : 14
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 )-2 x \left (t \right ) y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-y \left (t \right )+x \left (t \right ) y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.492 (sec). Leaf size: 99

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

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

✓ Mathematica. Time used: 0.13 (sec). Leaf size: 169

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

\begin{align*} y(t)&\to -W\left (-\frac {\exp \left (\frac {1}{2} \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (-\frac {e^{\frac {K[1]}{2}-c_1}}{\sqrt {K[1]}}\right )+1\right )}dK[1]\&\right ][2 t+c_2]-c_1\right )}{\sqrt {\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (-\frac {e^{\frac {K[1]}{2}-c_1}}{\sqrt {K[1]}}\right )+1\right )}dK[1]\&\right ][2 t+c_2]}}\right )\\ x(t)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (-\frac {e^{\frac {K[1]}{2}-c_1}}{\sqrt {K[1]}}\right )+1\right )}dK[1]\&\right ][2 t+c_2] \end{align*}

✗ Sympy

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