80.8.11 problem 15

Internal problem ID [21368]
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 : 15
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 )&=x \left (t \right )-4 x \left (t \right ) y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-2 y \left (t \right )+x \left (t \right ) y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.386 (sec). Leaf size: 77

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

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

✓ Mathematica. Time used: 0.094 (sec). Leaf size: 141

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

\begin{align*} y(t)&\to -\frac {1}{4} W\left (-\frac {4 \exp \left (\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (-\frac {4 e^{K[1]-c_1}}{K[1]^2}\right )+1\right )}dK[1]\&\right ][t+c_2]-c_1\right )}{\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (-\frac {4 e^{K[1]-c_1}}{K[1]^2}\right )+1\right )}dK[1]\&\right ][t+c_2]{}^2}\right )\\ x(t)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (-\frac {4 e^{K[1]-c_1}}{K[1]^2}\right )+1\right )}dK[1]\&\right ][t+c_2] \end{align*}

✗ Sympy

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