80.11.25 problem 26

Internal problem ID [21433]
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 12. Stability theory. Excercise 12.6 at page 270
Problem number : 26
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 )^{3}\\ \frac {d}{d t}y \left (t \right )&=-y \left (t \right )^{3} \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 47

ode:=[diff(x(t),t) = -x(t)^3, diff(y(t),t) = -y(t)^3]; 
dsolve(ode);
 

\begin{align*} \left \{x \left (t \right ) &= \frac {1}{\sqrt {2 t +c_2}}, x \left (t \right ) = -\frac {1}{\sqrt {2 t +c_2}}\right \} \\ \left \{y \left (t \right ) &= \frac {1}{\sqrt {2 t +c_1}}, y \left (t \right ) = -\frac {1}{\sqrt {2 t +c_1}}\right \} \\ \end{align*}

✓ Mathematica. Time used: 0.169 (sec). Leaf size: 226

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

\begin{align*} x(t)&\to -\frac {1}{\sqrt {2 t-2 c_1}}\\ y(t)&\to -\frac {1}{\sqrt {2 t-2 c_2}}\\ x(t)&\to -\frac {1}{\sqrt {2 t-2 c_1}}\\ y(t)&\to \frac {1}{\sqrt {2 t-2 c_2}}\\ x(t)&\to -\frac {1}{\sqrt {2 t-2 c_1}}\\ y(t)&\to 0\\ x(t)&\to \frac {1}{\sqrt {2 t-2 c_1}}\\ y(t)&\to -\frac {1}{\sqrt {2 t-2 c_2}}\\ x(t)&\to \frac {1}{\sqrt {2 t-2 c_1}}\\ y(t)&\to \frac {1}{\sqrt {2 t-2 c_2}}\\ x(t)&\to \frac {1}{\sqrt {2 t-2 c_1}}\\ y(t)&\to 0\\ x(t)&\to 0\\ y(t)&\to -\frac {1}{\sqrt {2 t-2 c_2}}\\ x(t)&\to 0\\ y(t)&\to \frac {1}{\sqrt {2 t-2 c_2}}\\ x(t)&\to 0\\ y(t)&\to 0 \end{align*}

✓ Sympy. Time used: 0.106 (sec). Leaf size: 66

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

\[ \left [ x{\left (t \right )} = \frac {1}{\sqrt {C_{1} + 2 C_{2} + 2 t}}, \ y{\left (t \right )} = - \frac {\sqrt {2} \sqrt {\frac {1}{C_{2} + t}}}{2}, \ x{\left (t \right )} = \frac {1}{\sqrt {C_{1} + 2 C_{2} + 2 t}}, \ y{\left (t \right )} = \frac {\sqrt {2} \sqrt {\frac {1}{C_{2} + t}}}{2}\right ] \]