76.30.1 problem 1

Internal problem ID [17818]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 7. Nonlinear Differential Equations and Stability. Section 7.1 (Autonomous Systems and Stability). Problems at page 464
Problem number : 1
Date solved : Monday, March 31, 2025 at 04:33:18 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.823 (sec). Leaf size: 93

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

\begin{align*} \left [\{x \left (t \right ) = 2\}, \left \{y \left (t \right ) &= -\frac {1}{4}+{\mathrm e}^{8 t} c_1\right \}\right ] \\ \left [\left \{x \left (t \right ) &= \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}\frac {4 \,{\mathrm e}^{\operatorname {RootOf}\left (-32 \,{\mathrm e}^{\textit {\_Z}} \ln \left (\textit {\_a} -2\right )-\ln \left (\frac {{\mathrm e}^{\textit {\_Z}}}{2}+\frac {9}{2}\right ) {\mathrm e}^{\textit {\_Z}}+3 \,{\mathrm e}^{\textit {\_Z}} c_1 +\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-16 \,{\mathrm e}^{\textit {\_Z}} \textit {\_a} +9\right )}}{9 \left (\textit {\_a} -2\right )}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 )-2}\right \}\right ] \\ \end{align*}

✓ Mathematica. Time used: 0.23 (sec). Leaf size: 112

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

\begin{align*} y(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {K[1]}{4 K[1]+1}dK[1]\&\right ]\left [\int _1^{x(t)}\frac {K[2]}{K[2]-2}dK[2]+c_1\right ] \\ \text {Solve}\left [\int _1^{x(t)}\frac {1}{(K[3]-2) \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {K[1]}{4 K[1]+1}dK[1]\&\right ]\left [c_1+\int _1^{K[3]}\frac {K[2]}{K[2]-2}dK[2]\right ]}dK[3]&=t+c_2,x(t)\right ] \\ \end{align*}

✗ Sympy

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

Timed Out