4.4.7 problem 9

Internal problem ID [1188]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Section 2.5. Page 88
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 04:27:49 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 47

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

\[ y = {\mathrm e}^{\operatorname {RootOf}\left (-\ln \left ({\mathrm e}^{\textit {\_Z}}-2\right ) {\mathrm e}^{\textit {\_Z}}+2 c_1 \,{\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+2 t \,{\mathrm e}^{\textit {\_Z}}+\ln \left ({\mathrm e}^{\textit {\_Z}}-2\right )-2 c_1 -\textit {\_Z} -2 t -2\right )}-1 \]

✓ Mathematica. Time used: 0.145 (sec). Leaf size: 51

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

\begin{align*} y(t)&\to \text {InverseFunction}\left [\frac {1}{\text {$\#$1}}+\frac {1}{2} \log (1-\text {$\#$1})-\frac {1}{2} \log (\text {$\#$1}+1)\&\right ][t+c_1]\\ y(t)&\to -1\\ y(t)&\to 0\\ y(t)&\to 1 \end{align*}

✓ Sympy. Time used: 0.253 (sec). Leaf size: 24

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

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