76.10.11 problem 11

Internal problem ID [17462]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.6 (A brief introduction to nonlinear systems). Problems at page 195
Problem number : 11
Date solved : Monday, March 31, 2025 at 04:14:21 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )^{2} y \left (t \right )-3 x \left (t \right )^{2}-4 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-2 x \left (t \right ) y \left (t \right )^{2}+6 x \left (t \right ) y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.242 (sec). Leaf size: 102

ode:=[diff(x(t),t) = 2*x(t)^2*y(t)-3*x(t)^2-4*y(t), diff(y(t),t) = -2*x(t)*y(t)^2+6*x(t)*y(t)]; 
dsolve(ode);
 

\begin{align*} \left \{x \left (t \right ) &= \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {9 \textit {\_a}^{4}+c_1 \,\textit {\_a}^{2}-18 \textit {\_a}^{2}-2 c_1 +36}}d \textit {\_a} +t +c_2 \right ), x \left (t \right ) = \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}-\frac {1}{\sqrt {9 \textit {\_a}^{4}+c_1 \,\textit {\_a}^{2}-18 \textit {\_a}^{2}-2 c_1 +36}}d \textit {\_a} +t +c_2 \right )\right \} \\ \left \{y \left (t \right ) &= \frac {-3 x \left (t \right )^{2}-\frac {d}{d t}x \left (t \right )}{-2 x \left (t \right )^{2}+4}\right \} \\ \end{align*}

✓ Mathematica. Time used: 0.602 (sec). Leaf size: 2113

ode={D[x[t],t]==2*x[t]^2*y[t]-3*x[t]^2-4*y[t],D[y[t],t]==-2*x[t]*y[t]^2+6*x[t]*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t)**2*y(t) + 3*x(t)**2 + 4*y(t) + Derivative(x(t), t),0),Eq(2*x(t)*y(t)**2 - 6*x(t)*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)