87.30.17 problem 17

Internal problem ID [23916]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 8. Nonlinear differential equations and systems. Exercise at page 310
Problem number : 17
Date solved : Sunday, October 12, 2025 at 05:55:17 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.213 (sec). Leaf size: 104

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

\begin{align*} \left [\{y \left (t \right ) = 0\}, \left \{x \left (t \right ) &= \frac {1}{-1+{\mathrm e}^{t} c_1}\right \}\right ] \\ \left [\left \{y \left (t \right ) &= \operatorname {RootOf}\left ({\mathrm e}^{-t}-3 \int _{}^{\textit {\_Z}}\frac {1}{\sqrt {12 \textit {\_f}^{4}-3 c_1 \textit {\_f}}}d \textit {\_f} +c_2 \right ) {\mathrm e}^{-t}, y \left (t \right ) = \operatorname {RootOf}\left ({\mathrm e}^{-t}+3 \int _{}^{\textit {\_Z}}\frac {1}{\sqrt {12 \textit {\_f}^{4}-3 c_1 \textit {\_f}}}d \textit {\_f} +c_2 \right ) {\mathrm e}^{-t}\right \}, \left \{x \left (t \right ) = \frac {\frac {d}{d t}y \left (t \right )+y \left (t \right )}{2 y \left (t \right )}\right \}\right ] \\ \end{align*}

✗ Mathematica

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

Not solved

✗ Sympy

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

Timed Out