82.8.27 problem 36-28 (a)

Internal problem ID [21898]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 36. Nonlinear differential equations. Page 1203
Problem number : 36-28 (a)
Date solved : Sunday, October 12, 2025 at 05:51:36 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.114 (sec). Leaf size: 68

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

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

✓ Mathematica. Time used: 0.089 (sec). Leaf size: 142

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

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

✗ Sympy

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