56.4.63 problem 60

Internal problem ID [8952]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 60
Date solved : Sunday, March 30, 2025 at 01:56:15 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=y \left (1-y^{2}\right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 29

ode:=diff(y(x),x) = y(x)*(1-y(x)^2); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {1}{\sqrt {{\mathrm e}^{-2 x} c_1 +1}} \\ y &= -\frac {1}{\sqrt {{\mathrm e}^{-2 x} c_1 +1}} \\ \end{align*}

✓ Mathematica. Time used: 0.731 (sec). Leaf size: 100

ode=D[y[x],x]==y[x]*(1-y[x]^2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {e^x}{\sqrt {e^{2 x}+e^{2 c_1}}} \\ y(x)\to \frac {e^x}{\sqrt {e^{2 x}+e^{2 c_1}}} \\ y(x)\to -1 \\ y(x)\to 0 \\ y(x)\to 1 \\ y(x)\to -\frac {e^x}{\sqrt {e^{2 x}}} \\ y(x)\to \frac {e^x}{\sqrt {e^{2 x}}} \\ \end{align*}

✓ Sympy. Time used: 1.096 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((y(x)**2 - 1)*y(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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